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PHYSICS OF PLASMAS

VOLUME 8, NUMBER 7

JULY 2001

A Landau fluid model for electromagnetic plasma microturbulence P. B. Snydera) General Atomics, P.O. Box 85608, San Diego, California 92186

G. W. Hammett Princeton Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, New Jersey 08543

共Received 11 December 2000; accepted 30 March 2001兲 A fluid model is developed for the description of microturbulence and transport in magnetized, long mean-free-path plasmas. The model incorporates both electrostatic and magnetic fluctuations, as well as finite Larmor radius and kinetic effects. Multispecies Landau fluid equations are derived from moments of the electromagnetic gyrokinetic equation, using fluid closures which model kinetic effects. A reduced description of electron dynamics, appropriate for the study of microturbulence on characteristic ion drift and Alfveń scales, is derived via an expansion in the electron to ion mass ratio. The reduced electron equations incorporate curvature, ⵜB, and linear and nonlinear E⫻B drift effects, needed to model the electron contribution to the drive and damping of ion gyroradius scale instabilities in tokamaks. The Landau fluid model is linearly benchmarked against gyrokinetic codes, and found to reproduce the toroidal finite beta ion temperature gradient and kinetic ballooning instabilities. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1374238兴 I. INTRODUCTION

which include magnetic fluctuations and nonadiabatic electrons. It should be noted that the collisionless ITG mode and the ion drift resonance driven KBM instability, considered in Sec. VI, both require kinetic effects not present in the standard Braginskii model for an accurate description. Here we develop an extension of earlier electrostatic gyrofluid models30,31 to incorporate magnetic fluctuations and nonadiabatic passing electron dynamics. 共This model can alternately be viewed as an extension of Waltz et al.14 to include more ion moments, the mirror force, different models of toroidal kinetic effects, and a numerically efficient reduced electron model.兲 A set of general multispecies electromagnetic gyrofluid equations are derived from velocity space moments of the nonlinear gyrokinetic equation in Sec. III. The moment hierarchy is truncated using a set of closures derived to model kinetic effects, including collisionless phase mixing due to parallel streaming and toroidal drifts, as well as linear and nonlinear finite-Larmor-radius 共FLR兲 effects. The general set of gyrofluid equations can be used to describe electron as well as ion dynamics. However, for many problems a more numerically efficient reduced model is appropriate for the electrons. The derivation of a reduced electron model which can be implemented in practical numerical simulations of electromagnetic ion drift and Alfveń scale turbulence is a key result of this paper. Section IV describes the physical motivation and mathematical derivation of the reduced electron equations. These reduced electron equations include the effects of electron temperature and density gradients, electron E⫻B motion, Landau damping, electron–ion collisions and the parallel electron currents which, along with parallel ion currents, give rise to the parallel magnetic potential. The system of equations is completed with the gyrokinetic Poisson equation and parallel Ampere’s Law in Sec. V. The Landau fluid system of equations is then benchmarked with linear gyrokinetic theory in Sec. VI. Linear growth rates and frequencies are compared

The development of an accurate and numerically efficient model of plasma microturbulence and transport in the kinetic, long mean-free-path regime characteristic of the core of magnetic fusion devices is a long standing challenge. Progress has been made via the development of the nonlinear gyrokinetic equation,1–4 and its numerical solution using direct,5,6 particle in cell,2,7–9 and ‘‘gyrofluid’’ 10–14 methods. Gyrofluid models take velocity space moments of the fivedimensional gyrokinetic equation to produce a reduced threedimensional fluid description. Kinetic effects are modeled via appropriately chosen fluid closures. Here we use the term ‘‘Landau fluid,’’ which emphasizes the use of fluid closures which model Landau damping, interchangeably with ‘‘gyrofluid,’’ which emphasizes that the fluid equations are moments of the gyrokinetic equation in gyrocenter space. The importance of incorporating magnetic fluctuations 共also called finite ␤ effects, where ␤ is the ratio of plasma pressure to magnetic pressure兲 in descriptions of ion gyroradius scale dynamics, has been identified by numerous authors. Magnetic fluctuations impact the growth rates of predominantly electrostatic linear instabilities, for example the finite ␤ stabilization of the collisionless toroidal ion temperature gradient 共ITG兲 mode,15,16 and introduce electromagnetic instabilities, such as the kinetic ballooning mode 共KBM兲.17–22 In addition, magnetic fluctuations are expected to significantly impact nonlinear dynamics and zonal flow generation.23,24 Linear and nonlinear electromagnetic effects are well documented in the extensive literature on collisional plasmas. The strong impact of magnetic fluctuations in collisional Braginskii25 simulations 共see Refs. 23, 26–29 and references therein for details on recent work in electromagnetic edge turbulence兲 motivates the development of models for the dynamics of kinetic, long mean-free-path plasmas a兲

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© 2001 American Institute of Physics

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Phys. Plasmas, Vol. 8, No. 7, July 2001

P. B. Snyder and G. W. Hammett

for both the toroidal finite ␤ collisionless ITG mode and the kinetic ballooning mode. Additional benchmarks and results from nonlinear toroidal turbulence simulations using the electromagnetic Landau fluid model developed here are presented in Ref. 32.

II. THE GYROKINETIC EQUATION

The starting point for the derivation of the fluid equations is the nonlinear electromagnetic gyrokinetic equation of Brizard,33 based upon earlier gyrokinetic work by many authors.1–4,34,35 The standard gyrokinetic ordering is invoked as follows:

␻ k 储 v ti e ␾ ␦ B F 1 ␳ i ⬃ ⬃ ⬃ ⬃␧Ⰶ1, ⬃ ⬃ ⍀i ⍀i T B F0 L

k⬜ ␳ i ⬃1,

共1兲

where ␻ is a characteristic frequency of the fluctuations, and k 储 and k⬜ are typical fluctuation wave numbers parallel and perpendicular to the equilibrium magnetic field. ⍀ i is the ion cyclotron frequency, v ti ⫽ 冑T i /m i is the ion thermal speed, and ␳ i ⫽ v ti /⍀ i is the thermal ion gyroradius. L is a typical equilibrium scale length, such as the density scale length L n ⫽⫺ⵜ(ln n0)⫺1, the temperature scale length L T ⫽⫺ⵜ(ln T0)⫺1, or the plasma minor radius (a) or major radius (R). T and B are typical equilibrium temperatures and magnetic fields, and F 0 is the equilibrium distribution. F 1 is the fluctuating distribution function, ␾ is the electrostatic potential 共which is assumed to have no equilibrium component兲, and ␦ B is the fluctuating component of the magnetic field. Gyrokinetics averages over the fast gyromotion of the particles around a strong magnetic field, reducing the kinetic equation from three to two velocity space dimensions, and leaving the magnetic moment ␮ as a rigorously conserved quantity. The gyrokinetic ordering takes advantage of the spatial anisotropy created by the strong magnetic field. Parallel to the field, particles can stream freely, and fluctuating wavelengths are long, k 储 L⬃1. Perpendicular to the field, particle motion is strongly restricted, and wavelengths scale with the gyroradius k⬜ ␳ i ⬃1. The fluctuating distribution function is ordered small compared to the equilibrium distribution, which here is taken as a Maxwellian. Nonetheless, perpendicular gradients of fluctuating quantities are the same order as perpendicular gradients of the equilibrium (k⬜ F 1 ⬃F 0 /L), and hence the perpendicular nonlinearities due to the E⫻B drift and field line bending are kept, while parallel nonlinearities are small, and are ordered out here. Brizard’s electromagnetic gyrokinetic equation can be written in the form

⳵F ⳵F ˙ •ⵜF⫹ v˙ 储 ⫹X ⫽C 共 F 兲 , ⳵t ⳵v储

共2兲

where F is the gyrocenter distribution function in the gyrocenter phase space coordinates (X, v 储 , ␮ , ␨ ). Within the gyrokinetic ordering ( ␻ Ⰶ⍀ i ), the gyrophase angle ␨ is effectively averaged over, and does not appear explicitly ( ⳵ F/ ⳵␨ ⫽0). The gyrocenter magnetic moment ␮ ⫽ v⬜2 /2B

⫹O(␧) is exactly conserved and enters the equations only as a parameter. An as yet undefined collision operator C(F) has been added to the right-hand side. Equation 共2兲 is solved through O(␧ 2 ) in the gyrokinetic ordering defined above. When ordering terms in the gyrokinetic equation, all frequencies are compared to ⍀ i , and all lengths to ␳ i . Hence, for example, ⳵ F/ ⳵ t⬃ ␻ F 1 is O(␧ 2 ), because ⳵ F 0 / ⳵ t⫽0, F 1 /F 0 ⬃␧, and ␻ /⍀ i ⬃␧. Any gradient operator acting on F 0 or B is O(␧) because ␳ i /L⬃␧. A parallel gradient on F 1 is O(␧ 2 ) because k 储 ␳ i ⬃␧. However, a perpendicular gradient acting on F 1 is O(␧) because ˙ is needed only to O(␧), k⬜ ␳ i ⬃1. Because ⵜF is O(␧), X while v˙ 储 must include terms through O(␧ 2 ). The fluctuating magnetic field ␦ B is described to lowest order in terms of a magnetic potential along the equilibrium field, ␦ B⫽ⵜ⫻A 储 bˆ, where bˆ is a unit vector along the equilibrium field. Note that A 储 and A⬜ are fluctuating quantities. The equilibrium magnetic field is denoted by B or Bbˆ, never as a magnetic potential. The perturbation along the equilibrium field ( ␦ B 储 ) is small for ␤ Ⰶ1, as can be seen from perpendicular force balance, and ␦ B 储 is neglected here. The gyrocenter velocity is then given by

冋

˙ ⫽ v 储 bˆ⫹ X

具 ␦ B⬜ 典 B

册

⫹vE ⫹vd ,

共3兲

where the angular brackets denote gyroangle averages. The first term on the right represents free streaming along the total magnetic field. The second term is the gyroaveraged E⫻B drift velocity, vE ⫽(c/B)bˆ⫻ⵜ 具 ␾ 典 . vd is the combined curvature and ⵜB drift velocity. In general, vd can be written vd ⫽ ⫽

v 2储

⍀

bˆ⫻ 共 bˆ•ⵜbˆ兲 ⫹

v 2储 ⫹ ␮ B

␮ bˆ⫻ⵜB ⍀

B⫻ⵜB⫹

⍀B 2

v 2储

⍀B 2

bˆ⫻ 共 ⵜ⫻B⫻B兲 .

共4兲

Using the equilibrium relations ⵜ p⫽(1/c)J⫻B and ⵜ⫻B ⫽(4 ␲ /c)J, this can be written vd ⫽

v 2储 ⫹ ␮ B

⍀B 2

B⫻ⵜB⫹

v 2储

⍀B 2

bˆ⫻ⵜp.

共5兲

The second term on the right is small for ␤ Ⰶ1,36 and is neglected here for simplicity and to maintain consistency with neglecting ␦ B 储 . A cancellation occurs between the ⵜ p term in vd and a finite ␦ B 储 term.17,37 Hence it does not improve accuracy to keep the ⵜ p term until ␦ B 储 has been fully included. The definition vd ⬟

v 2储 ⫹ ␮ B

⍀B 2

共6兲

B⫻ⵜB,

is used henceforth. The gyrocenter parallel acceleration can be written v˙ 储 ⫽⫺

冉

冊

e ⳵ 具 A 储典 e 具 ␦ B⬜ 典 bˆ⫹ •ⵜ 具 ␾ 典 ⫺ mc ⳵ t m B

冉

⫺ ␮ bˆ⫹

具 ␦ B⬜ 典 B

冊

•ⵜB⫹ v 储共 bˆ•ⵜbˆ兲 •vE .


共7兲


A Landau fluid model for electromagnetic plasma . . .

The first two terms on the right-hand side represent the total parallel electric field, which includes both a magnetic induction term ⫺1/c( ⳵ 具 A 储典 / ⳵ t), and an electrostatic term evaluated along the total magnetic field. The next term is the total mirror force, and the final term is important for phase space conservation.30,38 Using the definition ␦ B⫽ⵜ⫻A 储 bˆ, the term ␦ B⬜ can be written as follows:

␦ B⬜ ⫽bˆ⫻ 共 ␦ B⫻bˆ兲 ⫽⫺bˆ⫻ⵜA 储 ⫹bˆ⫻bˆ•ⵜbˆ A 储 ,

共8兲

or upon gyroaveraging

具 ␦ B⬜ 典 ⫽⫺bˆ⫻ⵜ 具 A 储典 ⫹bˆ⫻bˆ•ⵜbˆ 具 A 储典 .

共9兲

The second term on the right hand side is O(␧ ) and does not enter Eq. 共2兲 to the required order. The gyroangle averages are expressed in terms of a gyroaveraging operator J 0 as follows: 2

具 ␾ 典 ⫽J 0 共 ␣ 兲 ␾ ,

具 A 储典 ⫽J 0 共 ␣ 兲 A 储 ,

to refer to the single ion species unless otherwise noted. The full, normalized equations for the multispecies case are given in Sec. III E. Because k⬜ ␳ i ⬃1, finite-Larmor-radius 共FLR兲 effects must be accounted for, both in the moment equations and in the closures. In order to simplify the process of taking velocity space moments, all functions of velocity space 共F,J 0 , ␮ , v 储 , etc.兲 are moved to the same side of the spatial and temporal operators. The first two terms in Eq. 共11兲, B( ⳵ F/ ⳵ t)⫽( ⳵ / ⳵ t)FB and B v 储 bˆ•ⵜF⫽B•ⵜ(FB v 储 /B) are easily put in a form suitable for taking moments. The next three terms require modification. Noting that spatial derivatives are taken with ␮ and v 储 fixed, we can write for any field A ⵜJ 0 A⫽J 0 ⵜA⫹AⵜJ 0 ,

␣ ⬟⫺i

ⵜJ 0 共 ␣ 兲 ⫽

ⵜ⬜ ,

⍀i

冑2 ␮ B v ti

k⬜ ␳ i .

␾ ⬘⫽ ␾ ⫺

The operator J 0 , J 0共 ␣ 兲 ⫽

1 2␲ ⬁

⫽

0

1 1

兺 2 n⫽0 共 n! 兲

冉冊冉冑冊 i␣ 2

⳵F ⫹ 共 v 储 ˜b⫹vE ⫹vd 兲 •ⵜF ⳵t

冋

c vE⬘ ⫽ bˆ⫻ⵜJ 0 ␾ ⬘ , B

BvE⬘ •ⵜF⫽B

2n

2␮B 2⍀

2n

ⵜ⬜2n ,

共10兲

is a simple Bessel function in Fourier space. In real space, J 0 does not in general commute with other operators, and must be manipulated with care. J 0 operates only on the electrostatic potential ␾ and the parallel magnetic potential A 储 . Defining the unit vector along the total magnetic field ˜b⫽bˆ⫹( 具 ␦ B⬜ 典 /B) and the total parallel electric field ˜E 储 ˜ •ⵜJ 0 ␾ , the gyrokinetic equation ⫽⫺(1/c)( ⳵ / ⳵ t)J 0 A 储 ⫺b can be written

⫹

v储 A , c 储

共13兲

c v␾⬘ ⫽ bˆ⫻ⵜ ␾ ⬘ . 共14兲 B

Using Eqs. 共12兲 and 共13兲 d ␨ exp共 i ␣ cos ␨ 兲

兺 2 n⫽0 共 n! 兲 ⬁

⫽

冕

2␲

⳵J0 ␣ ⵜ ␣ ⫽J 1 共 ␣ 兲 ⵜB. ⳵␣ 2B

The term representing free streaming along the fluctuating magnetic field, ⫺ v 储 /B(bˆ⫻ⵜJ 0 A 储 ), can be combined with the E⫻B drift by introducing the operator notation:

or, in Fourier space

␣⫽

共12兲

where

where ␣ is the operator defined by

冑2 ␮ B

3201

册

e ⳵F ˜E ⫺ ␮ ˜b•ⵜB⫹ v 储共 bˆ•ⵜbˆ兲 •vE ⫽C 共 F 兲 . m 储 ⳵v储

III. ELECTROMAGNETIC GYROFLUID EQUATIONS

Gyrofluid equations are derived by taking velocity space moments of Eq. 共11兲, and implementing closures to model kinetic effects. For simplicity of notation, the derivation for a single ion species is presented here. The subscript i is omitted in this section, and all quantities 共v t ,⍀,T, etc.兲 are taken

冊

共15兲

The J 1 term above can be neglected as it is O(␧ 3 ) due to the presence of ␾ ⬘ , ⵜB, and ⵜF, each of which are O(␧). Noting that J 0 ⵜF⫽ⵜJ 0 F⫺( ␣ /2B)FJ 1 ⵜB, and introducing the operator notation i ␻ d⬟

v 2t

⍀B 2

共16兲

B⫻ⵜB•ⵜ,

allows us to write B

c bˆ⫻J 0 ⵜ ␾ ⬘ •ⵜF B ⫽B

c␣ c bˆ⫻ⵜ ␾ ⬘ •ⵜ 共 J 0 F 兲 ⫺F J bˆ⫻ⵜ ␾ ⬘ •ⵜB B 2B 1

⫽v␾⬘ •ⵜ 共 J 0 FB 兲 ⫹ 共11兲

冉

␣ c ␾ ⬘ ⵜB •ⵜF. bˆ⫻ J 0 ⵜ ␾ ⬘ ⫹J 1 B 2B

冉

冊

␣ e FB J 0 ⫹J 1 i ␻ d ␾ ⬘ . T 2

共17兲

Invoking the approximation outlined in Eqs. 共4兲–共6兲, and noting that i ␻ d B⫽0, the ⵜB and curvature drift term can be written Bvd •ⵜF⫽i ␻ d 关 FB 共 v 2储 ⫹ ␮ B 兲兴 .

共18兲

Turning now to the v˙ 储 ( ⳵ F/ ⳵ v 储 ) terms, we note first that all components of v˙ 储 except the lowest order mirror force ⫺ ␮ bˆ•ⵜB are O(␧ 2 ) and therefore involve only the equilibrium distribution, which is taken to be


3202


F 0⫽

n0

e 共 2 ␲ v 2t 兲 3/2

2

2

2

⫺ v 储 /2v t ⫺ ␮ B/ v t


The mirror force terms can be written

共19兲

.

⳵F ⳵F B 共 ⫺ ␮ bˆ•ⵜB 兲 ⫽⫺ ␮ B 2 bˆ•ⵜ ln B, ⳵v储 ⳵v储

The electric field terms can be written as follows to O(␧ 2 )

⳵F ⳵F0 ␮ J 共 ⵜB⫻B兲 •ⵜA 储 ␮ 共 bˆ⫻ⵜJ 0 A 储兲 •ⵜB⫽ ⳵v储 ⳵v储 B 0

⳵F e e ⳵F0 ⫺B bˆ•ⵜ 共 J 0 ␾ 兲 ⫽⫺ Bbˆ•ⵜ 共 J 0 ␾ 兲 m ⳵v储 m ⳵v储

冉

冊

⳵F0 ⳵F0 e e ⫽⫺ bˆ•ⵜ BJ 0 ␾ ⫹ J 0 ␾ m ⳵v储 m ⳵v储

冉

⫻B 1⫺

␮B v 2t

冊

bˆ•ⵜ ln B,

共20兲

e ⳵F0 ⳵F e 共 bˆ⫻ⵜJ 0 A 储兲 •ⵜJ 0 ␾ ⫽ 共 bˆ⫻J 0 ⵜA 储兲 •J 0 ⵜ ␾ ⳵v储 m m ⳵v储

共22兲

⫽⫺

⳵F0 e␮B BJ 0 i ␻ dA 储 , ⳵v储 cT

共23兲

where the J 1 ⵜB term has cancelled exactly. Finally, the phase space conservation term can be rewritten, omitting the finite ␤ component for consistency with the treatment of the curvature drift

⳵F ⳵F0 c B v 共 bˆ•ⵜbˆ兲 •vE ⫽⫺ B⫻ⵜB•ⵜJ 0 ␾ v ⳵v储储 ⳵v储储 B2

e ⳵F0 J J 共 bˆ⫻ⵜA 储兲 •ⵜ ␾ , m ⳵ v 储 0A 储 0 ␾

冋

共24兲

册

共21兲

e ⳵ 共 F BJ v 兲 ⫺F 0 BJ 0 i ␻ d ␾ . T ⳵v储 0 0 储共25兲

where the notation J 0A 储 and J 0 ␾ is used to indicate the field on which the Bessel function operator acts. All J 1 terms above have been dropped as they are O(␧ 3 ).

Combining all the above terms, and defining ⵜ 储 ⫽bˆ•ⵜ, the electromagnetic gyrokinetic equation can be written in the following cumbersome but useful form:

⫽

⫽⫺

⳵ e␾⬘ ␣ e␾⬘ e 1 FB⫹Bⵜ 储 F v 储 ⫹v␾⬘ •ⵜ 共 FBJ 0 兲 ⫹2FBJ 0 i ␻ d ⫹FBJ 1 i ␻ d ⫹ v 储 FBJ 0 i ␻ d A 储 ⫹ 2 i ␻ d 关 FB 共 v 2储 ⫹ ␮ B 兲兴 ⳵t T 2 T cT vt ⫺

冉

冊

冉

冉

冊

冊

⳵A储 ⳵F0 ⳵F0 ␮B e ⳵ e e e ⳵F0 F 0 BJ 0 ⫺ ⵜ储 BJ 0 ␾ ⫹ J 0 ␾ B 1⫺ 2 ⵜ 储 ln B⫹ J J mc ⳵ v 储 ⳵t m ⳵v储 m ⳵v储 m ⳵ v 储 0A 储 0 ␾ vt

⫻ 共 bˆ⫻ⵜA 储兲 •ⵜ ␾ ⫺ ␮ B 2

⳵F ⳵F0 e␮B ⳵ e ⵜ ln B⫺ BJ 0 i ␻ dA 储⫺ 共 FBJ 0 v 储兲 i ␻ d ␾ ⫽0. ⳵v储储 ⳵v储 cT ⳵v储 T

Nearly all terms with velocity space dependence are now grouped on the same side of spatial and temporal operators so that moments may easily be taken. The exception is the v 储 term which appears in ␾ ⬘ ⫽ ␾ ⫺( v 储 /c)A 储 and v␾⬘ ⫽(c/B)bˆ ⫻ⵜ ␾ ⬘ . However, v 储 commutes with J 0 , J 1 and all spatial operators, and may be easily moved to the appropriate place inside velocity space integrals. The collision operator C(F) has been omitted here. Collisions are considered in Sec. III D. Equation 共26兲 contains terms through O(␧ 2 ) in the gyrokinetic ordering. Assuming a time independent equilibrium distribution F 0 with gradients that scale as 1/L, only two first-order terms remain. These terms represent free streaming along the equilibrium field, and the lowest order mirror force. To first order, the equation can be written Bⵜ 储 F 0 v 储 ⫺ ␮ B 2

⳵F0 ⵜ ln B⫽0, ⳵v储储

共27兲

a condition which is satisfied exactly by the equilibrium Maxwellian

F 0 ⫽F M ⫽

n0 共 2 ␲ v 2t 兲 3/2

共26兲

2

2

2

e ⫺ v 储 /2v t ⫺ ␮ B/ v t .

This leaves only second-order terms in the equation. We furthermore divide the first-order distribution F 1 into two parts, F 1 ⬟˜f ⫹F 1nc . Here F 1nc is defined to be an equilibrium part of the distribution with no time dependence and gradients which scale as 1/L. It is further defined to be an exact solution of the equation Bⵜ 储 F 1nc v 储 ⫹ ⫽0.

1 v 2t

i ␻ d 关 F 0 B 共 v 2储 ⫹ ␮ B 兲兴 ⫺ ␮ B 2

⳵ F 1nc ⵜ ln B ⳵v储储共28兲

Note that the F 1nc contribution to all other terms is O(␧ 3 ) or higher and can be neglected. This removes all terms with no time dependence, and leaves us with an evolution equation for the fluctuating first-order distribution ˜f , containing only second-order terms which are either linear or quadratic in the fluctuating quantities ˜f , ␾, and A 储




冋

册

3203

⳵ e␾ eA 储 ␣ e␾ v v储 ˜f B⫹Bⵜ 储˜f v 储 ⫹v␾ •ⵜ 关共 F 0 ⫹˜f 兲 BJ 0 兴 ⫺vA •ⵜ 共 F 0 ⫹˜f 兲 B 储 J 0 ⫹2F 0 BJ 0 i ␻ d ⫺F 0 B J 0 i ␻ d ⫹F 0 BJ 1 i ␻ d 储 ⳵t c T c T 2 T ⫺F 0 B ⫹

冉

冉

冊

冊

eA 储 i ␻ d ⳵A储 e ⳵F0 ⳵F0 ␮B e ⳵F0 e v储 ␣ BJ 0 BJ 0 ␾ ⫹ J 0 ␾ B 1⫺ 2 ⵜ 储 ln B J i␻ ⫹ 2 关˜f B 共 v 2储 ⫹ ␮ B 兲兴 ⫺ ⫺ ⵜ储 c 12 d T mc ⳵ ⳵ t m ⳵ m ⳵ v v v vt vt 储储储

e ⳵F0 ⳵˜f ⳵F0 ␮B eA 储 ⳵ e␾ J 0A 储 J 0 ␾ 共 bˆ⫻ⵜA 储兲 •ⵜ ␾ ⫺ ␮ B 2 ⵜ 储 ln B⫺ BJ 0 i␻d ⫺ ⫽0. 共 F 0 BJ 0 v 储兲 i ␻ d m ⳵v储 ⳵v储 ⳵v储 c T ⳵v储 T

Terms containing ␾ and A 储 have been separated by defining v␾ ⫽(c/B)bˆ⫻ⵜ ␾ and vA 储 ⫽(c/B)bˆ⫻ⵜA 储 . Nonlinear terms enter through v␾ •ⵜ 关˜f BJ 0 兴 , vA •ⵜ 关˜f B( v 储 /c)J 0 兴 , and 储

e/m( ⳵ F 0 / ⳵ v 储 )J 0A 储 J 0 ␾ (bˆ⫻ⵜA 储 )•ⵜ ␾ . It is also possible to derive Eq. 共29兲 starting with the conservative form of the gyrokinetic equation. Making sure to include the second-order part of 具 ␦ B⬜ 典 from Eq. 共9兲, it is possible to prove Liouville’s theorem to the required order

⳵B* ⳵ ˙ 兴⫹ ⫹ⵜ• 关 B * X 关 B * v˙ 储兴 ⫽0, ⳵t ⳵v储

共30兲

where B * ⫽B⫹(mc/e) v 储 bˆ•ⵜ⫻bˆ contains the parallel velocity correction. The gyrokinetic equation can then be written

⳵ ⳵ ˙ 兴⫹ FB * ⫹ⵜ• 关 FB * X 关 FB * v˙ 储兴 ⫽0. ⳵t ⳵v储

共31兲

Again working within the context of the low ␤ approximation bˆ⫻(bˆ•ⵜbˆ)⫽(1/B 2 )B⫻ⵜB, and rearranging terms, one finds Eq. 共29兲 to second order as expected. A further check on Eq. 共29兲 is to calculate the linear nonadiabatic response in the local limit. Dividing the distribution into adiabatic and nonadiabatic pieces, ˜f ⫽g ⫺F 0 J 0 e ␾ /T 0 , linearizing, transforming, and taking the ⵜ 储 ln B⫽0 limit, we find the expected nonadiabatic distribution

冉

冊

␻⫺␻T e v储 * J0 ␾⫺ A储 , g⫽F 0 ␻ ⫺k 储 v 储 ⫺ ␻ d v T c

共32兲

where ␻ T ⫽ ␻ 关 1⫹ ␩ ( v 2储 /2v 2t ⫹ ␮ B/ v 2t ⫺3/2) 兴 , ␻ d v ⫽ ␻ d ( v 2储 * * ⫹ ␮ B)/ v 2t , and we have introduced the diamagnetic frequency i ␻ ⬟⫺(cT 0 /eBn 0 )ⵜn 0 •bˆ⫻ⵜ, and the ratio of * scale lengths ␩ ⫽L n /L T . A. The moment equations

Fluid moment equations can now be derived by taking velocity space moments of Eq. 共29兲. In this section a careful distinction is made between equilibrium and fluctuating components, and equilibrium quantities are written with a subscript 0. Both v t ⫽ 冑T 0 /m and ␳ i ⫽ v t /⍀ are defined in terms of equilibrium quantities. It should also be noted that because all terms in Eq. 共29兲 are O(␧ 2 ), only their lowest-order components need be kept, e.g., T→T 0 . Velocity space moments are often defined in terms of the total distribution function F. Here we again separate F into

共29兲

equilibrium and fluctuating components F⫽F 0 ⫹˜f , noting that F 1nc and its moments do not enter the equations to the required order and can be neglected. Velocity space moments of F 0 ⫽F M ⫽

n0

e 共 2 ␲ v 2t 兲 3/2

2

2

2

⫺ v 储 /2v t ⫺ ␮ B/ v t

,

are all well defined. We define the following moments of the fluctuating distribution: ñ ⫽

冕

˜p 储 ⫽m

˜f d 3 v ,

冕

n 0˜u 储 ⫽

˜f v 2储 d 3 v ,

冕

˜p⬜ ⫽m

冕冕

˜q 储 ⫽⫺3m v 2t n 0˜u 储 ⫹m ˜q⬜ ⫽⫺m v 2t n 0˜u 储 ⫹m ˜r 储 , 储 ⫽m

冕冕

˜r⬜,⬜ ⫽m

˜f v 4储 d 3 v ,

˜f v 储 d 3 v ,

冕

˜f B ␮ d 3 v ,

˜f v 3储 d 3 v ,

˜f B ␮ v 储 d 3 v ,

˜r 储 ,⬜ ⫽m

冕

˜f B ␮ v 2储 d 3 v ,

˜f B 2 ␮ 2 d 3 v ,

冕冕冕

˜s⬜,⬜ ⫽⫺2m v 4t n 0˜u 储 ⫹m

˜f B 2 ␮ 2 v 储 d 3 v ,

˜s 储 , 储 ⫽⫺15m v 4t n 0˜u 储 ⫹m

˜f v 5储 d 3 v ,

˜s 储 ,⬜ ⫽⫺3m v 4t n 0˜u 储 ⫹m

˜f B ␮ v 3储 d 3 v ,

where d 3 v ⫽2 ␲ d v 储 B d ␮ . The definitions of the q and s moments above have been chosen for consistency of notation with Beer.30 Each moment is coupled to higher moments through the terms in Eq. 共29兲 which contain factors of v 储 or ␮, including terms due to parallel free streaming, toroidal drift, FLR effects, and the mirror force. This moment hierarchy is truncated using closures described in the following sections in order to generate a useful set of equations. Taking integrals of Eq. 共29兲 of the form 2 ␲ 兰 d v 储 d ␮ v 储j ␮ k , and defining the shorthand 具 A 典 ⬟2 ␲ 兰 Ad v 储 Bd ␮ yields the following set of moment equations:


3204



⳵ñ n 0˜u 储 1 ⫹Bⵜ 储 ⫹v␾ •ⵜ 具 FJ 0 典 ⫺ vA 储 •ⵜ 具 F v 储 J 0 典 ⳵t B c ␣ 1 e␾ ⫹ F 0 2J 0 ⫹J 1 i␻d ⫹ i ␻ 共 ˜p ⫹p ˜ 兲 ⫽0, 2 T0 T0 d 储 ⬜

冓冉

n0

冊冔

⳵ ˜p 储 ˜q 储 ⫹3 p 0˜u 储 m ⫹Bⵜ 储 ⫹mv␾ •ⵜ 具 F v 2储 J 0 典 ⫺ vA 储 •ⵜ 具 F v 3储 J 0 典 ⳵t B c

冓冉

⫹m F 0 v 2储 2J 0 ⫹J 1 共33兲

⳵˜u 储 ˜p 储 1 ⫹Bⵜ 储 ⫹v␾ •ⵜ 具 F v 储 J 0 典 ⫺ vA 储 •ⵜ 具 F v 2储 J 0 典 ⳵t mB c

冓冉

⫺ F 0 v 2储 J 0 ⫹J 1

␣ 2

冊冔

i␻d

冓冉

␮B e ␾ F 0 J 0 1⫺ 2 m vt

⫻ⵜA 储 •ⵜ ␾ ⫹

冊冔

eA 储 1 ⫹ i␻ cT 0 T 0 d

ⵜ 储 ln B⫺

冋

册

⫺

⫻i ␻ d

共34兲

冊冔

i␻d

冓冉

i ␻ 共˜s 储储 ⫹s ˜ 储 ,⬜ ⫹18m v 4t n 0˜u 储兲 ⫹ v 2t d ,

⫺

3e eA 储 ⫽0, 具 F 0 v 2储 J 0A 储 J 0 ␾ 典 bˆ⫻ⵜA 储 •ⵜ ␾ ⫹3r˜ 储 ,⬜ ⵜ 储 ln B⫹3mB 具 F 0 ␮ v 2储 J 0 典 i ␻ d B cT

eA 储 cT 0

冊冔共36兲

冊冔

冓

共37兲

冊冔冓冉冊冔冉

⳵ ˜r 储 ,⬜ ␣ mB vA 储 •ⵜ 具 F v 2储 ␮ J 0 典 ⫺mB F 0 v 2储 ␮ J 0 ⫹J 1 共 ˜q⬜ ⫹p 0˜u 储兲 ⫹B 2 ⵜ 储 2 ⫹mBv␾ •ⵜ 具 F v 储 ␮ J 0 典 ⫺ ⳵t B c 2 i ␻ 共˜s 储 ⫹s ˜ ⫹5m v 4t n 0˜u 储兲 ⫹ v 2t d ,⬜ ⬜,⬜

冓冉

共35兲

⳵A储 ␮B 3e ⫹3eⵜ 储具 F 0 v 2储 J 0 典 ␾ ⫺3e F 0 v 2储 1⫺ 2 J 0 ␾ ⵜ 储 ln B 具 F 0 v 2储 J 0 典 c ⳵t vt

⫹

1

e␾ ⫽0, T0

e␾ 1 ⫹ 2 i ␻ d 共˜r 储 ,⬜ ⫹r ˜⬜,⬜ 兲 ⫽0, T0 vt

冓冉

⫹

e␾ 1 ⫹ i ␻ 共˜r ⫹r ˜ 兲 T 0 v 2t d 储 , 储储 ,⬜

␣ mB v •ⵜ 具 F ␮ v 储 J 0 典 ⫹mB F 0 ␮ 2J 0 ⫹J 1 c A储 2

⳵ ˜r 储 , 储 ␣ m ⫹mv␾ •ⵜ 具 F v 3储 J 0 典 ⫺ vA 储 •ⵜ 具 F v 4储 J 0 典 ⫺m F 0 v 4储 J 0 ⫹J 1 共 ˜q ⫹3p 0˜u 储兲 ⫹Bⵜ 储 ⳵t 储 B c 2 1

i␻d

⳵ ˜p⬜ 1 ⫹B 2 ⵜ 储 2 共 ˜q⬜ ⫹ p 0˜u 储兲 ⫹mBv␾ •ⵜ 具 F ␮ J 0 典 ⳵t B

e 具 F J J 典 bˆ mB 0 0A 储 0 ␾

˜p⬜ eA 储 ⵜ 储 ln B⫹ 具 F 0 ␮ BJ 0 典 i ␻ d ⫽0, m cT 0

冊冔

⫹2 共 ˜q⬜ ⫹ p 0˜u 储兲 ⵜ 储 ln B⫹2m 具 F 0 v 2储 J 0 典 i ␻ d

e ⳵A储 e ⫻ 共 ˜q 储 ⫹q ˜ ⬜ ⫹4p 0˜u 储兲 ⫹ 具 F 0 J 0 典 ⫹ ⵜ 储具 F 0J 0典 ␾ mc ⳵ t m ⫺

␣ 2

i␻d

eA 储 ct 0

⳵A储 ␮B eB ⫹eBⵜ 储具 F 0 ␮ J 0 典 ␾ ⫺eB F 0 ␮ 1⫺ 2 J 0 ␾ ⵜ 储 ln B 具 F 0␮ J 0典 c ⳵t vt

˜⬜,⬜ ⵜ 储 ln B⫹mB 2 具 F 0 ␮ 2 J 0 典 i ␻ d ⫺e 具 F 0 ␮ J 0A 储 J 0 ␾ 典 bˆ⫻ⵜA 储 •ⵜ ␾ ⫹r

B. Finite Larmor radius terms

Closures are developed for the finite Larmor radius terms appearing in Eqs. 共33兲–共38兲, using the techniques of Dorland39 as adapted to the toroidal case by Beer.30 We choose to evolve ion moments in guiding center space rather than real space in order to better describe both linear and nonlinear FLR effects, including the Bakshi–Linsker effect.40 Nonetheless, our FLR terms, when expanded, contain higher velocity space moments and these must be carefully closed to properly model kinetic behavior. Turning first to the Maxwellian FLR terms, we must 2i j j close terms of the forms 具 F 0 v 2i 储 ␮ J 0 典 and 具 F 0 v 储 ␮ J 1 ␣ 典 , where i⫽0,1,2 and j⫽0,1,2. Note that purely Maxwellian FLR terms with odd powers of v 储 vanish identically, as F M is even in v 储 , while J 0 and J 1 ␣ are independent of v 储 . The FLR closures are chosen in careful consideration of the entire system of equations. It is the combination of J 0

eA 储 ⫽0. cT 0

共38兲

terms from the E⫻B and vA 储 terms with the J 0 terms in Poisson’s equation and Ampere’s Law which motivates the basic approximation 具 J 0 典 ⬇ 具 J 20 典 1/2⬇⌫ 0 (b) 1/2, where b ⫽k⬜2 ␳ 2i . Following and extending Ref. 39, we choose

具 F 0 J 0 典 ⫽n 0 ⌫ 1/2 0 ,

共39兲

具 F 0 J 0 v 2储典 ⫽n 0 v 2t ⌫ 1/2 0 ,

共40兲

具 F 0J 0␮ 典 ⫽

n 0 v 2t ⳵ v 2t ˆ2 共 b⌫ 1/2 共 2⌫ 1/2 0 兲⫽ 0 ⫹ⵜ⬜ 兲 , B ⳵b 2B

具 F 0 J 0 v 4储典 ⫽3n 0 v 4t ⌫ 1/2 0 , 具 F 0 J 0 v 2储 ␮ 典 ⫽

n 0 v 4t ⳵ v 4t ˆ2 ⫽ 兲共 b⌫ 1/2 共 2⌫ 1/2 0 0 ⫹ⵜ⬜ 兲 , B ⳵b 2B

共41兲共42兲共43兲



具 F 0J 0␮ 典 ⬇ 2

⫽

v 4t

B

2

v 4t

B2

冋

b


⳵2 ⳵ 共 b⌫ 1/2 共 b⌫ 1/2 0 兲 ⫹2b 0 兲 ⳵b2 ⳵b

册

ˆ2 92 共 2⌫ 1/2 0 ⫹ⵜ⬜ ⫹ⵜ⬜ 兲 .

共44兲

9 2 are deˆ 2 and ⵜ The modified Laplacian operators ⵜ ⬜ ⬜ fined as follows:

⳵ ⌫ 1/2 1 2 ˆ ⌽⫽b 0 ␾ , ⵜ 2 ⬜ ⳵b 9 2 ⌽⫽b ⵜ ⬜

共45兲

⳵2 共 b⌫ 1/2 0 兲␾, ⳵b2

共46兲

where the notation ⌽⫽⌫ 1/2 0 ␾ has been introduced for the gyroaveraged electrostatic potential. The analogous notation A储 ⫽⌫ 1/2 0 A 储 is used for the gyroaveraged magnetic potential. The J 1 terms are evaluated following Ref. 30, using the following trick:

具 FJ 1 ␣ 典 ⬇⫺

⳵ ⳵␨

冏

␨ ⫽1

具 FJ 0 共 ␨ ␣ 兲典 .

共47兲

eraging terms are acted on by a perpendicular gradient operator, requiring that gradients of both fluctuating and equilibrium quantities be kept. In considering these terms, we redefine b⬟(1/⍀) 冑(T⬜ /m), in terms of the total perpendicular temperature T⬜ , which contains both an equilibrium part (T 0 , as the equilibrium is assumed isotropic兲 and a fluc˜ ⬜ ⫺T 0ñ )/n 0 . The gradient of b is then tuating part, ˜T⬜ ⫽(p calculated as follows: ⵜb⫽

b 2b ˜⬜ 兲⫺ ⵜB. 共 ⵜT 0 ⫹ⵜT T0 B

v␾ •ⵜ 具 J 0 F 典 ⫽v␾ •ⵜ 关 n⌫ 1/2 0 共 b 兲兴 ,

冏

⳵ ⌫ 1/2 0 2 2 ˆ2, ⫽⫺ v 2t ⵜ 具 F 0 J 1 v 储 ␣ 典 ⬇⫺2 v t b ⬜ ⳵b ⳵ 具 F 0 J 1 ␮␣ 典 ⬇⫺ ⳵␨

冏

共49兲

v␾ •ⵜ 具 J 0 F 典 ⫽⫺n 0 i ␻ ⌫ 1/2 0

冉

冊

具 F 0 J 1 v 4储 ␣ 典 ⬇⫺6 v 4t b

⳵ ⌫ 1/2 0 ˆ2, ⫽⫺3 v 4t ⵜ ⬜ ⳵b

v4

t 9 2 具 F 0 J 1 v 2储 ␮␣ 典 ⬇⫺2 B ⵜ ⬜.

共50兲共51兲共52兲

The Maxwellian terms which contain more than one factor of J 0 are closed analogously 1/2 1/2 ⌫ 0␾ , 具 F 0 J 0A 储 J 0 ␾ 典 ⫽n 0 ⌫ 0A 储

共53兲

1/2 1/2 ⌫ 0␾ , 具 F 0 v 2储 J 0A J 0 ␾ 典 ⫽n 0 v 2t ⌫ 0A

共54兲

储

具 F 0 ␮ J 0A J 0 ␾ 典 ⫽ 储

储

v 2t

2B

1/2 ˆ2 ˆ2 关共 2⌫ 1/2 0 ⫹ⵜ⬜ 兲 A 储 ⫹ 共 2⌫ 0 ⫹ⵜ⬜ 兲 ␾ 兴 ,

⫹n 0 ⵜˆ ⬜2 i ␻ d

e␾ n0 e⌽ ⫺ ␩ i ⵜˆ ⬜2 i ␻ * T0 T0 2

e␾ ⫹N L . T0

共58兲

Nonlinear terms arise both from ñ and b, and can be written N L ⫽v⌽ •ⵜn ˜⫹

n0 ˆ 2 v 兴 •ⵜT ˜⬜ . 关ⵜ 2T 0 ⬜ ⌽

共59兲

To account for the vA 储 •ⵜ and v␾ •ⵜ terms with higher powers of v 储 and ␮, we note that the linear terms from Eq. 共58兲 can be generalized as follows:

v 2t ⳵ 关 T 具F J 共 ␨ ␣ 兲典兴 B ⳵ T⬜ ⬜ 0 0 ␨ ⫽1

⳵ ⌫ 1/2 v 2t ⳵ v 2t 2 0 2 9 , ⫽⫺2 b ⫽⫺2 ⵜ B ⳵b ⳵b B ⬜

共57兲

˜ . Introducing the diamagwhere n is the total density, n 0 ⫹n netic frequency i ␻ ⬟⫺(cT 0 /eBn 0 )ⵜn 0 •bˆ⫻ⵜ, and ratio * ␩ i ⫽L n /L T , where L T is the scale length of the equilibrium temperature, this leads to three linear terms

*

⳵ ⌫ 1/2 0 1/2 2 ⌫ 0 共 ␨ b 兲 ⫽⫺2b ⫽⫺ⵜˆ ⬜2 , ⳵ b ␨ ⫽1 共48兲

共56兲

Closing these FLR terms analogously to Eqs. 共39兲–共44兲 leads to, for example,

Again using 具 FJ 0 典 ⬇⌫ 1/2 0 yields

⳵ 具 F 0 J 1 ␣ 典 ⬇⫺ ⳵␨

3205

共55兲

where the subscript ␾ or A 储 again designates the field on which the operator acts. These closures can be thought of in terms of separate expansions of the two Bessel function operators, through first order in b, so that no cross term enters. The v␾ •ⵜ 具 FJ 0 ¯ 典 and vA 储 •ⵜ 具 FJ 0 ¯ 典 terms introduce two additional complications. These terms contain both the Maxwellian and the perturbed distribution, and the gyroav-

v␾ •ⵜn 0 g 共 b 兲 ⫽⫺n 0 g 共 b 兲 i ␻ ⫹2n 0 b

e␾ ⳵g e␾ ⫺n 0 ␩ i b i␻ * T0 * ⳵b T0

⳵g e␾ i␻d . ⳵b T0

共60兲

The treatment of the nonlinear terms is somewhat more subtle, as these can involve higher moments which are not evolved. Following Ref. 39, and introducing the notation N L (x) for the nonlinear terms generated by v␾ •ⵜ 具 FJ 0 x 典 N L 共 v 储兲 ⫽n 0 v⌽ •ⵜu ˜ 储⫹

1 ˆ 2 v 兴 •ⵜq ˜⬜ , 关ⵜ 2T 0 ⬜ ⌽

共61兲

˜ 储⫹ N L 共 m v 2储兲 ⫽v⌽ •ⵜp

n0 2 ˆ v 兴 •ⵜT ˜⬜ , 关ⵜ 2 ⬜ ⌽

共62兲

N L 共 mB ␮ 兲 ⫽v⌽ •ⵜp ˜ ⬜ ⫹ 21 关 ⵜˆ ⬜2 v⌽ 兴 •ⵜq ˜⬜ ,

共63兲

˜ 储 ⫹3 p 0 v⌽ •ⵜu ˜ 储 ⫹ 23 关 ⵜˆ ⬜2 v⌽ 兴 •ⵜq ˜⬜ , N L 共 m v 3储兲 ⫽v⌽ •ⵜq ˜ ⬜ ⫹ p 0 v⌽ •ⵜu ˜ 储⫹ N L 共 mB v 储 ␮ 兲 ⫽v⌽ •ⵜq

共64兲

p0 2 ˆ v 兴 •ⵜu ˜储关ⵜ 2 ⬜ ⌽

⫹ 12 关 ⵜˆ ⬜2 v⌽ 兴 •ⵜq ˜ ⬜ ⫹ 关 ⵜ9 ⬜2 v⌽ 兴 •ⵜq ˜⬜ .

共65兲


3206



The vA•ⵜ ¯ nonlinear terms are closed identically to the above with the substitution ⌽→A储 . However, the vA•ⵜ ¯ terms in the ˜q 储 and ˜q⬜ equations contain higher moments which are closed using results from the next section. To simplify the equations, we introduce the following normalization. Time, parallel lengths, and perpendicular

ˆ ,Aˆ 储 ,nˆ ,uˆ ,pˆ ,qˆ ,rˆ ,sˆ 兲 ⫽ 共␾

lengths are normalized to v t /L n , L n and ␳ i , respectively: 共 ˆt ,kˆ 储 ,kˆ⬜ 兲 ⫽

冉

冊

tvt ,k L ,k ␳ , Ln 储 n ⬜ i

共66兲

and the fluctuating quantities are normalized as follows:

冉

冊

L n e ␾ A 储 ñ ˜u ˜p ˜q ˜r ˜s , . , , , 2, 3, 4, ␳ i T 0 ␳ i B n 0 v t n 0 m v t n 0 m v t n 0 m v t n 0 m v 5t

Normalized quantities appear on the left. The caret designating a normalized quantity is dropped for simplicity of notation. Note that these normalizations mesh with the gyrokinetic ordering such that all characteristic drift scales are O(1). Because ␤ is formally taken to be O(1), all shear Alfveń scales are O(1) as well. C. Closures

Closures must be introduced for the highest moments, r and s, in order to have a complete and useful set of gyrofluid equations. The terms requiring closure divide naturally into three categories, the parallel terms ⵜ 储 r 储 , 储 and ⵜ 储 r 储 ,⬜ , the toroidal terms ␻ d (r 储 , 储 ⫹r 储 ,⬜ ), ␻ d (r 储 ,⬜ ⫹r⬜,⬜ ), ␻ d (s 储 , 储 ⫹s 储 ,⬜ ), and ␻ d (s 储 ,⬜ ⫹s⬜,⬜ ), and the mirroring terms r 储 , 储 ⵜ 储 ln B, r 储 ,⬜ ⵜ 储 ln B, and r⬜,⬜ ⵜ 储 ln B. Following the work of Beer,30 we separately treat each group of terms, making closure approximations that accurately model the physical processes that each set of terms represents. 1. Parallel Landau closures

Closures which provide an accurate model of linear Lan˜ 储 r 储储 and dau damping are chosen for the parallel terms, ⵜ , ˜ ˜ ⵜ 储 r 储 ,⬜ , where we have introduced the notation ⵜ 储 ⫽ⵜ 储 ⫺vA 储 •ⵜ⫽ⵜ 储 ⫺bˆ⫻ⵜA储 •ⵜ. Landau damping along the magnetic field occurs due to the velocity dependence of the k 储 v 储 term in the kinetic equation. Components with different k 储 stream along the field at different velocities, causing moments of F to phase mix away. As an illustration, consider the one-dimensional kinetic equation

⳵f ⳵f ⫹ v 储 ⫽ ␦ 共 t 兲 f 0 共 z, v 兲 , ⳵t ⳵z

共68兲

where f 0 provides the initial condition. The solution to this simple equation f (z, v ,t)⫽ f 0 (z⫺ v t, v )H(t), provides Green’s functions which can be used to solve more general problems with additional source terms, such as the electric field ⫺(e/m)E 储 ( ⳵ F M / ⳵ v ). Consider an initial condition with a small single harmonic perturbation f 0 ⫽(n 0 ⫹n 1 e ikz )F M ( v ). The general solution is just 关 n 0 ⫹n 1 e ik(z⫺ v t) 兴 , which simply oscillates in time at ␻ ⫽k v and does not damp. However, upon taking velocity space moments, the velocity integration introduces mixing of the phases as follows:

n 共 z,t 兲 ⫽

冕

f d v ⫽n 0 ⫹n 1

共67兲

e ikz

冑

2 ␲ v 2t

冕

d v e ⫺ik v t e ⫺ v

2 /(2 2 ) vt

. 共69兲

2 ⫺k 2 v t t 2 /2

decays with a The perturbed density n 1 ⫽n 1(t⫽0) e Gaussian time dependence. This decay due to linear Landau damping is not captured by a simple fluid model with a finite number of moments, and hence it must be accounted for in the fluid closure if it is to be included in a fluid model. A number of different ‘‘Landau closures’’ which model linear Landau damping in fluid models have been developed.41–46 Here the four moment model of Refs. 13, 30, and 44 is employed. This closure accurately models linear kinetic response functions, conserves energy, and takes a simple, frequency independent form in Fourier space, allowing for easy implementation in nonlinear initial value simulations. The introduction of electromagnetic effects does not significantly alter the process of deriving linear Landau closures. Response functions are simply written in terms of the total E 储 rather than ␾. In Ref. 47, Landau closures are derived for the general electromagnetic case with both parallel and perpendicular magnetic fluctuations. Here we consider only perpendicular fluctuations, hence the magnitude of the fluctuating field ˜B is zero to first order in the perturbation. The general response functions and closures are given in Sec. IV of Ref. 47. Here we take the B 1 ⫽0 limit of that result, for the case in which the equilibrium distribution is isotropic. In this limit the result is identical to the earlier result of Ref. 12 r 储 , 储 ⫽3 共 2 p 储 ⫺n 兲 ⫹c 储 T 储 ⫺&D 储 r 储 ,⬜ ⫽ p 储 ⫹ p⬜ ⫺n⫺&D⬜

i 兩 k 储兩 q 储 , k储

i 兩 k 储兩 q⬜ , k储

共70兲共71兲

where c 储 ⫽(32⫺9 ␲ )/(3 ␲ ⫺8), D 储 ⫽2 冑␲ /(3 ␲ ⫺8), and D⬜ ⫽ 冑␲ /2. Note that here and elsewhere the dissipative terms in the closure (⬃ 兩 k 储兩 /k 储 ) are written in their Fourier space form for conciseness. In configuration space these terms are convolution integrals. Because the dissipative part of the closure above 共the 兩 k 储兩 /k 储 terms兲 is written in terms of moments with no equilibrium component, the fluctuating field makes no contribution to the linear Landau closures. Hence the linear Landau closure is equally accurate in the electrostatic and electro-




3207

magnetic cases. However, there is an additional nonlinear Landau damping term due to A 储 which is discussed in Sec. VII of Ref. 47. This and other nonlinear Landau damping mechanisms are not accounted for in the fluid closures given here. As discussed in Ref. 39, nonlinear phase-mixing may be important at large amplitudes or large k⬜ ␳ , and extensions to include some nonlinear phase mixing effects are of interest for future work.

r⬜,⬜ ⫽4 p⬜ ⫺2n.

共78兲

2. Toroidal closures

The velocity dependence of the ⵜB and curvature drifts also introduces phase mixing. This process is modeled using toroidal closures of Beer,30 which include dissipative pieces proportional to 兩 ␻ d 兩 / ␻ d . Beer’s closures include both Maxwellian parts and dissipative pieces derived by careful fitting with all parts of the kinetic toroidal response function, and can be written in the following form: r 储 , 储 ⫹r 储 ,⬜ ⫽7p 储 ⫹p⬜ ⫺4n⫺2i

兩 ␻ d兩共 ␯ T ⫹ ␯ 2 T⬜ 兲 , ␻d 1 储

共72兲

r 储 ,⬜ ⫹r⬜,⬜ ⫽p 储 ⫹5 p⬜ ⫺3n⫺2i

兩 ␻ d兩共 ␯ T ⫹ ␯ 4 T⬜ 兲 , ␻d 3 储

共73兲

s 储 , 储 ⫹s 储 ,⬜ ⫽⫺i

兩 ␻ d兩共 ␯ u ⫹ ␯ 6 q 储 ⫹ ␯ 7 q⬜ 兲 , ␻d 5 储

s 储 ,⬜ ⫹s⬜,⬜ ⫽⫺i

兩 ␻ d兩共 ␯ u ⫹ ␯ 9 q 储 ⫹ ␯ 10q⬜ 兲 , ␻d 8 储

共74兲

共75兲

where the complex coefficients take the form ␯ ⫽ ␯ r ⫹i ␯ i 兩 ␻ d 兩 / ␻ d . The coefficients chosen are, in the form ( ␯ r , ␯ i ), ␯ 1 ⫽(2.019,⫺1.620), ␯ 2 ⫽(0.433,1.018), ␯ 3 ⫽(⫺0.256,1.487), ␯ 4 ⫽(⫺0.070,⫺1.382), ␯ 5 ⫽(⫺8.927, 12.649), ␯ 6 ⫽(8.094,12.638), ␯ 7 ⫽(13.720,5.139), ␯ 8 ⫽(3.368,⫺8.110), ␯ 9 ⫽(1.974,⫺1.984), ␯ 10⫽(8.269, 2.06). As shown in Figs. 2.1 and 2.2 of Ref. 30, these closures provide a good fit to the linear toroidal response functions, including a reasonable model of the toroidal branch cut at ␻ / ␻ d ⫽⫺k 2储 v 2t /4␻ 2d . As noted in Ref. 30, this set of toroidal closures accurately models the fast linear collisionless damping of zonal flows for t⬍qR/ v ti 冑⑀ , but does not account for the residual undamped component of the zonal flow noted by Rosenbluth and Hinton.48 Efforts to incorporate this residual flow and other neoclassical effects into a new toroidal closure are ongoing.49 3. Mirroring closures

The mirroring terms r 储 , 储 ⵜ 储 ln B, r 储 ,⬜ ⵜ 储 ln B, and r⬜,⬜ ⵜ 储 ln B incorporate trapped ion effects and magnetic pumping. However, they do not introduce new dissipative processes and hence they are closed with simple Maxwellian closures, again following Ref. 30 r 储 , 储 ⫽6p 储 ⫺3n,

共76兲

r 储 ,⬜ ⫽p 储 ⫹p⬜ ⫺n,

共77兲

Taken together, the closure approximations provide models of linear and nonlinear FLR effects, as well as parallel phase mixing, drift resonance, and trapped particle effects. The accuracy of these closures is tested extensively through linear benchmarks with kinetic theory given in Sec. VI and Ref. 32. D. Ion collisions

Ion–ion collisions are modeled with a simple particle, momentum and energy conserving Bohm–Gross–Krook 共BGK兲 operator50 C 共 F j 兲 ⫽⫺

兺k ␯ jk共 F j ⫺F M jk 兲 ,

共79兲

where j and k are species indices, and ␯ jk is the collision rate of species j with species k. Collisions cause the distribution to relax to a shifted Maxwellian with the appropriate total (equilibrium⫹fluctuating) momentum and energy. Upon linearizing, the single species operator can be written

再冋

C 共 F 兲 ⫽⫺ ␯ ii F 1 ⫺ n⫹u 储

冉

3 v储 v2 ⫹T ⫺ vt 2 v 2t 2

冊册冎

F0 ,

共80兲

where n, u 储 , and T⫽(T 储 ⫹2T⬜ )/3 are normalized fluctuating moments, and v 2 ⫽ v 2储 ⫹4B 2 ␮ 2 . Ion–electron collisions are negligible due to the smallness of the electron–ion mass ratio. Electron–ion collisions are considered in Sec. IV C. E. Final gyrofluid equations

Incorporating the parallel, toroidal, and mirror term closures defined above, and including moments of the ion–ion collision operator, yields the final set of single species electromagnetic gyrofluid equations. The derivation in the previous sections has focused on a single ion species for simplicity. In general, tokamak plasmas contain multiple ion species, as well as electrons. In some cases, such as the deuterium–tritium plasmas used in fusion experiments, the bulk plasma may contain more than one dominant ion species. In addition, impurity ions are expected to play an important role, especially near the plasma edge. The extension to multiple species is reasonably straightforward. A separate set of gyrofluid equations is solved for each species j, noting that charge e, mass m, and the equilibrium moments 共n 0 , T 0 兲 and scale lengths are functions of the species j. Here each species is normalized to its own n 0 , v t , etc., but one ion species is chosen as a reference. The reference species is designated with the subscript i, and the following dimensionless constants are introduced, ␶ j ⫽T 0 j /T 0i , v j ⫽ v t j / v ti , and ␳ˆ j ⫽ ␳ j / ␳ i . Z is the ratio of the species charge to the unit charge, Z⫽e j / 兩 e 兩 , and the reference species i is assumed to have Z⫽1. ␩ j is the usual ratio of scale lengths ␩ j ⫽L n j /L T j . The basic macroscopic length is taken to be the electron density scale length L ne , and the following nor-


3208



malized scale length is defined for each ion species, Lˆ n j ⫽L n j /L ne . The multispecies equations are then written as follows:

dn

˜储 ⫹ v j Bⵜ

dt

冉

⫺ 1⫹

u储 B

⫹

冉

␩j ˆ

ⵜ⬜2

2

1 2

冊

冊

ˆ 2 v •ⵜT ⫺ v ⵜ ⬜ j ⬜ ⌽

冉

i␻

冉

1 2

dt

˜储 ⫹ v j Bⵜ

⫹

v jZ

␶j

冉

p储

冊

冊

B

冉

⫹

˜ⵜ 储 ⌽⫹

⫹ p⬜ ⫹

2

⫹ 4⫹

ˆ 2⌽ Z ⵜ ⬜

␶j

2

⳵t

冊

d p⬜

冉

1

dt

2

⫹ p⬜ ⫺p 储 ⫹

⫺ 1⫹

ˆ2 ⵜ ⬜

i␻

␶j

冉

2

冊

ˆ 2 v •ⵜT ⵜ ⬜ ⬜ ⌽

冋冉冊册

2

2

Lˆ n j

dq 储

v j ⵜ 储 ln B⫹i ␻ d ␳ˆ j v j 共 q 储 ⫹q⬜ ⫹4u 储兲

册

冉

1 2

ˆ 2 i␻ ⵜ ⬜ 2

9 2 ⌽⫺ ⵜ ⬜

2

i␻

2

Lˆ n j

*⌽

i ␻ d ⌽⫹i ␻ d ␳ˆ j v j 共 7 p 储 ⫹ p⬜ ⫺4n 兲

q⬜ ⫹u 储 B

2

⫹

ˆ2 ⵜ ⬜ 2

共83兲

冋册 1 2

ˆ 2 v •ⵜp ⫹ 关 ⵜ9 2 v 兴 •ⵜT ⵜ ⬜ ⬜ ⬜ ⌽ ⬜ ⌽

冉

⫹ ␩ j 1⫹

ˆ2 ⵜ ⬜ 2

⫹ⵜ9 ⬜2

冊册

i␻

*⌽

Lˆ n j

冊

共84兲

˜ 储 T 储 ⫹&D 储 v 兩 k 储兩 q 储 ⫹ 共 3⫹c 储兲v j ⵜ j ⫹i ␻ d ␳ˆ j v j 共 ⫺3q 储 ⫺3q⬜ ⫹6u 储兲 ⫹ 共 3⫹c 储兲 ␩ j

i␻

* v j A储

Lˆ n j

⫹ 兩 ␻ d ␳ˆ j v j 兩共 ␯ 5 u 储 ⫹ ␯ 6 q 储 ⫹ ␯ 7 q⬜ 兲 ⫽⫺ ␯ s q 储 ,

共85兲

ˆ 2 v •ⵜu 储 ⫹ 共 ⵜ9 2 v 兲 •ⵜq ⫺ v 共 ⵜ9 2 ⫺ 21 ⵜˆ 2 兲 v •ⵜT ⵜ ⬜ j ⬜ ⬜ ⌽ ⬜ ⌽ ⬜ ⬜ A

* v j A储 ⫹

Lˆ n j

ˆ 2⌽ ⵜ ⬜

ˆ2 ⵜ ⬜

⫹2 兩 ␻ d ␳ˆ j v j 兩共 ␯ 3 T 储 ⫹ ␯ 4 T⬜ 兲 ⫽ 31 ␯ s 共 p 储 ⫺ p⬜ 兲 ,

* v j A储

共82兲

Z

1

⫹ 共 3⫹ 23 ⵜˆ ⬜2 ⫹ⵜ9 ⬜2 兲 i ␻ d ⌽⫹i ␻ d ␳ˆ j v j 共 5 p⬜ ⫹ p 储 ⫺3n 兲

dt

9 2 兲共 1⫹ ␩ 兲 ⫹ ␩ j 共 1⫹ⵜ j ⬜

ˆ2 ⵜ ⬜

˜储 ⫹ v jB 2ⵜ

冋

冊

冋冉冊册

⫹ 1⫹ ␩ j 1⫹

˜ 储 T ⫹&D v 兩 k 储兩 q ⫹ ⫹v jⵜ ⬜ ⬜ j ⬜

冋冋

dt

ˆ 2 v •ⵜT ⵜ ⬜ ⬜ A

⫽0,

dq⬜

冉

⫺ v j 关 12 ⵜˆ ⬜2 vA兴 •ⵜ 共 q⬜ ⫹u 储兲

冊

␶j

B

⫹

⫹2 兩 ␻ d ␳ˆ j v j 兩共 ␯ 1 T 储 ⫹ ␯ 2 T⬜ 兲 ⫽⫺ 32 ␯ s 共 p 储 ⫺ p⬜ 兲 ,

共81兲

ˆ 2 v •ⵜq ⫺ v ⵜ ⬜ j ⬜ ⌽

v j Z ⳵ A储

q 储 ⫹3u 储

冉冊

ˆ 2 v •ⵜq ⵜ ⬜ ⬜ A

1 2 ˆ * ⌽⫹ 2⫹ ⵜ ⬜ i ␻ d⌽ 2 Lˆ n j

1

dt

˜储 ⫹ v j Bⵜ

⫹2 v j 共 q⬜ ⫹u 储兲 ⵜ 储 ln B⫺ 1⫹ ␩ j 1⫹

⫹i ␻ d ␳ˆ j v j 共 p 储 ⫹ p⬜ 兲 ⫽0,

du 储

dp储

冊册

冉

Z v j ⵜˆ ⬜2 dA储

␶j

2

dt

冊

˜ 储 ⌽⫺i ␻ ␳ˆ v A储 ⫹i ␻ ␳ˆ v 共 ⫺q 储 ⫺q ⫹u 储兲 ⫹ⵜ d j j d j j ⬜

v j ⵜ 储 ln B⫹ 兩 ␻ d ␳ˆ j v j 兩共 ␯ 8 u 储 ⫹ ␯ 9 q 储 ⫹ ␯ 10q⬜ 兲 ⫽⫺ ␯ s q⬜ ,

共86兲

where d/dt⫽( ⳵ / ⳵ t)⫹v⌽ •ⵜ. We emphasize that in the above equations, the fluid quantities (n,u 储 ,p 储 ,p⬜ ,...) are all for species j, and an implied j subscript as been dropped. For clarity, we explicitly write here their relation to the physical quantities 共 n,u 储 ,p 储 ,p⬜ ,T 储 ,T⬜ ,q 储 ,q⬜ 兲 ⫽

冉

where 0 subscripts refer to equilibrium values and 1 subscripts refer to unnormalized fluctuating values. The gyroavˆ 2 , and ⵜ 9 2 , which act only on the eraging operators ⌫ 1/2 , ⵜ 0

⬜

冊

L ne n 1 j u 1 j p 储1 j p⬜1 j T 储 1 j T⬜1 j q 储1 j q⬜1 j , , , , , , , , ␳ i n 0 j v t j n 0 j m j v 2t j n 0 j m j v 2t j T 0 j T 0 j n 0 j m j v 3t j n 0 j m j v 3t j

共 ⌽,A储兲 ⫽

冉

共87兲

冊

L ne e v t 1/2 ⌫ A . ⌫ 1/2 0 ␾, ␳ i T 0i c 0 储

共88兲

⬜

fields ␾ and A 储 , are also species dependent through their implied argument b j , which in Fourier space is k⬜2 ␳ 2j . Since ⌽ and A are gyroaveraged quantities, they also implicitly depend on species j, via

IV. THE ELECTRON LANDAU FLUID EQUATIONS

Electron dynamics can be described by the full set of gyrofluid equations in the previous section. However, the full set of ion and electron equations will then contain widely



separated spatial and temporal scales which make fully resolved explicit numerical simulation challenging. For many problems, a reduced set of equations, describing fluctuations on a smaller range of scales is appropriate. Here we will develop a reduced electron model appropriate for the description of fluctuations on the ion drift and shear Alfveń scales often associated with tokamak plasma microturbulence. An analytic expansion in the electron to ion mass ratio is constructed, allowing electron dynamics on the characteristic ion and Alfveń scales to be treated explicitly, while the fast electron transit time scale and the small spatial scales associated with the electron gyroradius and the electron skin depth are removed from the set of fluid equations to be solved numerically. 共This is not to suggest that microturbulence on ␳ e and c/ ␻ pe scales does not exist or is not important in some phenomena. When these scales are important, the full electron gyrofluid equations, or another appropriate physics model should be used.兲 The resulting electromagnetic electron Landau fluid equations include the effects of electron temperature and density gradients, electron E⫻B motion, Landau damping, electron–ion collisions, and the parallel electron currents which, along with parallel ion currents, give rise to the parallel magnetic potential. The equations given here focus on the dynamics of the passing electrons. Developing an electromagnetic model of trapped electron dynamics analogous to the electrostatic model of Beer30 is left as an important piece of future work. A. Analytic expansion in the electron mass ratio

We invoke an analytic expansion in the electron–ion mass ratio, similar to the technique employed by Kadomtsev and Pogutse.51 This expansion removes the small electron gyroradius scale and the fast electron transit time scale from the equations, leaving an efficient model appropriate for the study of turbulence on ion and shear Alfveń scales. A lowest order model, containing no finite electron mass terms, will be derived first. This simple model will then be extended to include higher order dissipative terms in Sec. IV C. 1. Electron FLR and transit

To efficiently study fluctuation scales on the order of the ion gyroradius, we employ a subsidiary formal ordering in the smallness of the electron–ion mass ratio in order to remove electron finite Larmor radius terms. The gyroaveraging operator J 0 can be expanded 1⫹k⬜2 ␳ 2e ⫹ ¯ . In the gyrokinetic ordering employed here, k⬜ ␳ i ⬃1 the first electron FLR term is O(m e /m i ). We introduce the subsidiary ordering parameter ␦ ⬃ 冑m e /m i and note that electron FLR effects first enter at O( ␦ 2 ). The small electron mass also implies a fast electron thermal speed ( v te Ⰷ v ti ), and rapid electron streaming along the magnetic field. The speed of this streaming motion along the field introduces a Courant constraint on the size of the time step which can be used in an explicit numerical simulation. Adding electron parallel dynamics to a simulation which previously modeled only ions reduces this time step constraint by a factor of 冑T e m i /T i m e ⬃60 for a deuterium fusion


3209

plasma. This is a severe numerical burden, though perhaps one that it may be possible to contemplate bearing in the near future, as computational power continues to increase. Imposing the mass ratio ordering 冑m e /m i ⬃ ␦ Ⰶ1, allows the fast electron transit motion to be analytically removed. 2. General ordering

The use of the electron–ion mass ratio as an ordering parameter has a long history in plasma physics. It has been invoked in many forms of the magnetohydrodynamic equations as well as in the more detailed equations of Kadomtsev and Pogutse,51 and in many other fluid and simplified kinetic formulations. In the context of gyrokinetics, the mass ratio expansion has generally been used to justify the neglect of electron FLR terms, and treatment of electron dynamics with the drift kinetic equation. Here we wish to consistently apply the ordering m e /m i ⬃O( ␦ 2 ) to all terms in the drift fluid equations. The fundamental assumption is that the fluctuating scales of interest are those typical of ion thermal, drift and gyro-motion, and those of shear Alfveń waves. Length and time scales associated with electron thermal and gyromotion are taken to be small. For a typical perpendicular wave number k⬜ , we impose the following ordering: k⬜⫺1 ⬃ ␳ i ⬃c/ ␻ pi Ⰷ ␳ e ,c/ ␻ pe ,

共89兲

where ␻ p is the plasma frequency. The lengths on the left are independent of the electron mass, while the two lengths on the right are proportional to 冑m e . Note that the skin depth c/ ␻ p j can be written as ␳ j 冑2/␤ j for the single species case, where the species ␤ j ⫽8 ␲ n 0 j T 0 j /B 2 . Formally taking ␤ ⬃O(1), the above ordering of lengths follows directly from 冑m e /m i ⬃O( ␦ ). For a typical fluctuation frequency ␻ we choose the ordering

␻ ⬃k 储 v ti ⬃ ␻ ⬃ ␻ Di ⬃ ␻ De ⬃k 储 c s ⬃k 储 v A Ⰶk 储 v te ⬃ ␻ ETG , * 共90兲 where ␻ D is the curvature and ⵜB drift frequency, c s ⫽ 冑T 0e /m i is the cold ion sound speed, and v A ⫽B/ 冑4 ␲ n 0 m i is the Alfveń speed. We define ␻ ETG to be a frequency characteristic of the electron temperature gradient 共ETG兲 mode. These short wavelength modes typically have k ␪ ⬃1/␳ e , and hence ␻ ETG⬃ 冑m i /m e ␻ , where ␻ is * * the diamagnetic frequency taken with k ␪ ␳ i ⬃1. The quantities on the left are independent of m e while those on the right . are proportional to m ⫺1/2 e The desired time and length scale orderings above follow directly from m e /m i ⬃O( ␦ 2 ) and ␤ j ⬃O(1). The constraints on the validity of this expansion are found through inspection of Eqs. 共89兲 and 共90兲. The separation of scales between the shear Alfveń frequency and the electron transit frequency 共and equivalently between ␳ i and the electron skin depth兲 requires ␤ e Ⰷ2m e /m i . In fusion relevant plasmas, this condition is generally satisfied everywhere except very near the plasma edge. Another constraint is provided by the condition ␻ Ⰶk 储 v te . Using a typical ballooning k 储 ⬇1/qR,

*


3210



and ␻ ⬇k ␪ ␳ i v ti /L n , this requires k ␪ ␳ i 冑m e /m i ⰆL n /qR. * For large k ␪ ␳ i ⬇1, this condition can break down in the edge region where q is often large, while L n can become rather short. B. Derivation of the electron equations

The formal expansion in mass ratio can now be used to derive a set of equations which describe electron dynamics consistent with the time and space scale orderings described above. All fluctuations, including those in the electron moments ñ e , ˜u 储 e , ˜p e , etc., are taken to occur on the ion/Alfveń scales. It is thus convenient to normalize fluctuating electron moments to the ion quantities, v ti and m i , so that a consistent ordering is easily maintained.52 The fluctuating electron moments are normalized as follows: 共 nˆ e ,uˆ e ,pˆ e ,qˆ e ,rˆ e ,sˆ e 兲

⫽

冉

冊

˜p e ˜q e ˜r e ˜s e L n ñ e ˜u e , , , 2 , 3 , 4 , ␳ i n 0 v ti n 0 m i v ti n 0 m i v ti n 0 m i v ti n 0 m i v 5ti 共91兲

where the normalized quantities on the left are all O(1). In the general multiple ion species case, the quantities m i and v ti above refer to the reference ion species, as in Sec. III E. This normalization differs from that employed in Sec. III E, where each species’ moments are normalized to its own mass and thermal velocity. Lengths, times, and the fields ␾ and A 储 are normalized as in the ion equations. The unsubscripted normalized operators are again defined in terms of the Z⫽1 ion charge (e), the reference ion species mass (m i ) and temperature (T 0i ), and the equilibrium electron density (n 0 ) and density scale length (L n ) ˆ ⫽⫺ i␻

*

L n cT 0i ⵜn 0 •bˆ⫻ⵜ, v ti eBn 0

L n cT 0i ˆ d⫽ B⫻ⵜB•ⵜ. i␻ v ti eB 3

共92兲共93兲

The normalized electron density equation is u储 ⳵ne ˜ 储 e ⫺i ␻ ␾ ⫹vE •ⵜn e ⫹Bⵜ * ⳵t B ⫹i ␻ d 共 2 ␾ ⫺2n e / ␶ ⫺T 储 e ⫺T⬜ e 兲 ⫽0,

共94兲

where the carets on normalized quantities have been dropped ˜ 储 ⫽ⵜ 储 ⫺bˆ⫻ⵜA 储 for conciseness of notation. The notation ⵜ •ⵜ has been employed. Note that no factors of m e appear in the above equation, and all terms are of the same order. The momentum equation can be written p储 A储 m e ⳵ u 储e m e ˜ 储 e ⫹ 共 1⫹ ␩ 兲 i ␻ ⫹ vE •ⵜu 储 e ⫹Bⵜ e * ␶ mi ⳵t mi B

⳵A储 me ˜ 储 ␾ ⫹p ⵜ 储 ln B ⫹ i ␻ d 共 q 储 e ⫹q⬜ e ⫹4u 储 e / ␶ 兲 ⫺ ⫺ⵜ ⬜e mi ⳵t ⫽0.

共95兲

The electron inertia term, which is associated with the electron skin depth, and the curvature and ⵜB drift terms are both small by a factor of m e /m i ⬃ ␦ 2 . Neglecting these higher-order terms, and expanding the pressure, noting that p 储 e ⫽T 储 e ⫹n e / ␶ because of the normalization to ion temperature, the momentum equation can be recast as a time evolution equation for the magnetic potential

⳵A储 A储 1 ˜ 储 ␾ ⫺ ˜ⵜ 储 n ⫺ⵜ ˜ 储 T 储 ⫺ 共 1⫹ ␩ 兲 i ␻ ⫹ⵜ e e e * ⳵t ␶ ␶ ⫹ 共 T 储 e ⫺T⬜ e 兲 ⵜ 储 ln B⫽0.

共96兲

The equations for T 储 e and T⬜ e needed to complete the above set come from the q 储 e and q⬜ e moment equations. The p 储 e and p⬜ e moment equations provide information on the next order evolution of the temperature fluctuations. The q 储 e and q⬜ e equations contain the higher moments r e and s e which are closed as in Sec. III C. However, the electron closure terms are not in general O(1). Consider, for example, the Maxwellian closure for the moment r 储 , 储 e . This closure is derived by taking the first-order fluctuating part of the generalized Maxwellian result r 储 , 储 e ⫽3 p 2储 /m e n e . The e factor of 1/m e insures that this term is O( ␦ ⫺2 ). In the normalized units r 储 , 储 e →6

mi mi T ⫹3 2 n e , ␶ m e 储e ␶ me

共97兲

and similarly for r 储 ,⬜ e and r⬜,⬜ e . The Landau damping portion of the closure is smaller than the Maxwellian part by 冑m e /m i , and is neglected here, though it is reconsidered in Sec. IV C. Before normalizing or substituting in the closures, the q 储 e equation can be written to lowest order ˜储 Bⵜ

˜r 储储 e B

˜储 ⫺3T 0e Bⵜ

冉

⫹3 ˜r 储⬜ e ⫺

˜p 储 e m eB

⫹3 ␩ e

冊

2 n 0 T 0e

me

i␻

eA 储 * cT 0i

T 0i ˜p ⵜ ln B⫽0, m e ⬜e 储

共98兲

where the d/dt and ␻ d terms again drop out, as they are higher order in m e /m i . Substituting the Maxwellian closures, normalizing and simplifying gives ˜ 储 T 储 ⫹ ␩ i ␻ A 储 / ␶ ⫽0. ⵜ e e

共99兲 * The second term on the left is the gradient of the equilibrium temperature T 0e along the perturbed field, bˆ⫻ⵜA 储 •ⵜT 0e , or ˜ 储 T , as T equivalently the gradient along the total field ⵜ 0e 0e is constant along the equilibrium field. Equation 共99兲 can thus be written in the more physically intuitive form ˜ 储共 T 储 ⫹T 兲 ⫽ ⵜ 0e e

1 共 B ⫹B1 兲 •ⵜ 共 T 储 e ⫹T 0e 兲 ⫽0. B 0

共100兲

Quite simply, the total temperature is constant along the total magnetic field including fluctuations. This result is expected from our ordering of the velocities v ti , v A Ⰶ v te . The speeds




of the microturbulence being evolved are all slow compared to v te , and furthermore the Alfveń speed at which the magnetic field fluctuates is also much less than the electron thermal speed. Hence as the field fluctuates across the equilibrium temperature gradient, the electrons are able to almost instantaneously re-thermalize, leaving no electron temperature gradient along the total field. Note that this condition is quite different from the occasionally employed closures ⵜ 储 ˜T e ⫽0, or ˜T e ⫽0, both of which fail to properly account for the magnetic fluctuations across the equilibrium temperature gradient, and lead to errors when ␩ e is finite. Turning now to the q⬜ e moment equation, and again inserting Maxwellian closures, normalizing, and keeping only the dominant terms, the equation becomes ˜ 储T ⫹ ⵜ ⬜e

␩ ei ␻ A 储 * ⫹ 共 T⬜ ⫺T 储兲 ⵜ 储 ln B⫽0. e e ␶

共101兲

Again the second term is simply the derivative along the perturbed field of the equilibrium temperature (T 0e ). A mirror force term enters as well. Equations 共99兲 and 共101兲 can be recast by defining T e ⫽(T 储 e ⫹T⬜ e )/2 and ␦ T e ⫽(T⬜ e ⫺T 储 e ). Note that once Eq. 共99兲 has been substituted into the momentum equation, the temperature enters the momentum equation only as a mirroring term ␦ T e ⵜ 储 ln B, and enters the density equation only as ⫺i ␻ d T e . The equations for T e and ␦ T e are ˜ 储T ⫹ ⵜ e

␩ ei ␻ A 储 ␦ T e * ⫹ ⵜ ln B⫽0, ␶ 2 储

共102兲

˜ 储 ⫹ⵜ 储 ln B 兲 ␦ T ⫽0. 共ⵜ e

共103兲

In either the small mirror force limit (ⵜ 储 ln B→0) or the high collisionality limit ( ␦ T e →0), the above equations reduce to ˜ 储 T ⫽⫺ ␩ i ␻ A 储 / ␶ . Because the model only describes ⵜ e e * passing electrons, we employ this simple limit. The full set of normalized electron equations is then

冉

冊

u储 ⳵ne n ˜ 储 e ⫺i ␻ ␾ ⫹2i ␻ ␾ ⫺ e ⫺T ⫽0, ⫹vE •ⵜn e ⫹Bⵜ d e * ⳵t B ␶ 共104兲 1 1 ⳵A储 ˜ 储 ␾ ⫺ ˜ⵜ 储 n ⫺ i ␻ A 储 ⫽0, ⫹ⵜ e ⳵t ␶ ␶ * ˜ 储 T ⫽⫺ ⵜ e

␩e i␻ A储 , * ␶

共105兲共106兲

where the ␻ d T e term in Eq. 共104兲 is evaluated by numerically inverting Eq. 共106兲. It is assumed that any fluctuating component of T e which is constant on a field line does not contribute significantly to the ␻ d T e term. We emphasize that, provided an appropriate numerical inversion of Eq. 共106兲 can be achieved, no separate closure approximations are required for nonlinear terms. In this lowest order expansion in m e /m i , nonlinear closure terms such as those associated with nonlinear Landau damping drop out naturally, and no separate assumptions about the smallness of nonlinear terms are needed.

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The above equations provide a simple description of electromagnetic electron dynamics on shear Alfveń and ion scales. While only two moment equations need be solved, the physics content of a full six moment model has been incorporated to lowest order in m e /m i . Though the model is simple, it represents a substantial improvement over the adiabatic electron models (n e / ␶ ⫽ ␾ ⫺ 具 ␾ 典 surface) that have been used to describe the passing electrons in many previous gyrofluid and gyrokinetic particle simulations. In addition to finite-␤ effects and Alfveń wave dynamics, the above model also incorporates electron E ⫻B, curvature, and ⵜB drift motion, as well as the E⫻B nonlinearity and nonlinear terms due to magnetic flutter. The accuracy of this model in describing both finite-␤ ion drift waves and shear Alfveń waves is gauged in Sec. VI with a series of linear benchmarks. Furthermore, the numerical challenge of resolving short electron space and time scales has been entirely removed. The electron mass appears nowhere in Eqs. 共104兲–共106兲 or in the normalizations 关Eq. 共91兲兴, and it is apparent that the electron scales, 共␳ e , c/ ␻ pe , k 储 v te , ␻ ETG兲, all of which contain the electron mass, have been successfully removed from the equations which are numerically evolved. It can be shown, in a proof analogous to that of Cowley,53 that this electron model preserves magnetic flux surfaces. C. Electron collisions and Landau damping

One consequence of keeping only the lowest order terms in the mass ratio expansion is the absence of any damping mechanism in the electron channel. It is well known that damping terms, even when linearly small, can significantly impact the nonlinear dynamics of an otherwise dissipationless system. While the gyrofluid system has dissipation through ion collisions and ion Landau damping, it is expected that damping in the electron channel may play an important role as well. The dominant electron damping mechanisms are expected to be Landau damping and pitch angle scattering collisions with ions. These effects are introduced by extending the mass ratio expansion to include terms of O( 冑m e /m i ). Equations 共94兲 and 共96兲 remain unchanged, but additional terms are introduced into the closures for T⬜ e and T 储 e . The ⫽T⬜(0) ⫽T (0) lowest order T (0) 储 e is given by Eq. 共106兲. The full e e can expressions for the first order corrections T⬜(1) and T (1) 储 e e be derived from Eqs. 共81兲–共86兲. Here the ␻ , ␻ d →0 limit is * taken, leaving only the correction due to Landau damping along the field ⵜ 储 T (1) ⫽ 储 e

冑␶

␲ me 兩k 兩u , 2 m i 储储e

共107兲

and T⬜(1) ⫽0, where the operator 兩 k 储兩 is again written in its e Fourier space form for conciseness. A more precise description of finite electron mass effects is possible either by using and T⬜(1) , or by employing the the full expressions for T (1) 储 e e full six moment equation set 关Eqs. 共81兲–共86兲兴 to describe the electrons. As in the general multispecies case, discussed in


3212



Sec. III C 1, nonlinear phase mixing effects are not incorporated in this closure. The range of resonant electron velocities ⌬ v 储 ⬃ ␻ /k 储 Ⰶ v te is fairly small in core plasmas, and one might worry about nonlinear particle trapping causing electron Landau damping to turn off. However, even a small amount of collisions can be important for such narrow resonances. The rate of scattering out of the resonant region is ␯ ei v 2e /(⌬ v 储 ) 2 , and for a wide range of core parameters this is large compared to relevant linear or nonlinear rates even though ␯ ei is small. While one would thus expect linear Landau damping to hold, nonlinear kinetic effects can be subtle, and comparisons between the simulation results using this reduced electron model and fully kinetic calculations including collisions will be interesting future work. Electron–ion collisions are modeled with a Lorentz pitch angle scattering operator. Adding this operator to the righthand side of the drift kinetic equation and taking moments leads to the following collision term in the normalized electron momentum equation: ⫺ ␯ ei

me 共 u ⫺u 储 i 兲 , m i 储e

共108兲

where ␯ ei is the effective scattering rate, normalized to , this term is ordered ␯ ei ⬃ ␦ ⫺1 v ti /L n . Because ␯ ei ⬃m ⫺1/2 e so that the collision term enters at O( ␦ ). This caveat allows a formally consistent ordering in the mass ratio. It is recognized that the collision term may be smaller than other neglected terms. The collision term is kept to assess the impact of this damping mechanism in the electron channel. Including the pitch angle scattering model and the firstorder temperature correction 关Eq. 共107兲兴 in Eqs. 共94兲 and 共96兲, in the limit of small mirror force, yields the following set of electron equations:

冉

冊

u储 ⳵ne n ˜ 储 e ⫺i ␻ ␾ ⫹2i ␻ ␾ ⫺ e ⫺T ⫽0, ⫹vE •ⵜn e ⫹Bⵜ d e * ⳵t B ␶ 共109兲

⳵A储 1 1 ˜ 储 ␾ ⫺ ˜ⵜ 储 n ⫺ i ␻ A 储 ⫺ ⫹ⵜ e ⳵t ␶ ␶ * ⫽ ␯ ei ˜ 储 T ⫽⫺ ⵜ e

冑

drifts, parallel ion flow, and an improved Landau damping model which properly phase-mixes E⫻B driven perturbations. The model can be reduced to the familiar adiabatic response in the appropriate limits. Taking the limits ␤ →0, which implies A 储 →0 from Eq. 共114兲, and m e /m i →0, Eq. 共110兲 reduces to the adiabatic electron response ⵜ 储 ( ␾ ⫺n e / ␶ )⫽0; or, with the appropriate choice of constants, n e ⫽ ␶ ( ␾ ⫺ 具 ␾ 典 ). The adiabatic response can also be derived in the formal limit k 储 →⬁. Upon neglect of the ‘‘small scale’’ effects associated with the ⵜ p term in the momentum equation 共here these are ˜ 储 n and i ␻ A 储 terms兲, and in the limit m /m →0, Eq. the ⵜ e e i * 共110兲 reduces to the parallel ideal magnetohydrodynamic ˜ 储 ␾ ⫽0. Including 共MHD兲 Ohm’s Law E 储 ⫽⫺( ⳵ A 储 / ⳵ t)⫺ⵜ the collisional term gives the parallel resistive MHD Ohm’s ˜ 储 n ⫺i ␻ A 储 ) terms gives a Law. Adding the ⫺1/␶ (ⵜ e * version of the extended MHD Ohm’s Law appropriate for ␻ Ⰶk 储 v te . V. POISSON’S EQUATION AND AMPERE’S LAW

The system of equations is completed using the gyrokinetic Poisson’s equation and Ampere’s Law. In the limit of small Debye length, k␭ D Ⰶ1, the gyrokinetic Poisson’s equation becomes a quasineutrality constraint2 n e ⫽n ¯ i ⫺ 共 1⫺⌫ 0 兲 ␾ ,

where ¯n i is the gyrophase independent part of the real space ion density. The (1⫺⌫ 0 ) ␾ term, often called the polarization density, arises from the gyrophase dependent part of the distribution function, and accounts for the difference between guiding center density and ion particle density. Following Beer,30 the transformation from gyrocenter to real space is accomplished with the simple Pade´ approximation: ¯n i ⫽

␲ me 兩k 兩u 2 ␶ m i 储储e

me 共 u ⫺u 储 i 兲 , m i 储e

共110兲

␩e i␻ A储 . * ␶

共111兲

The w d T (1) term has been neglected, and the ⵜ 储 T (1) intro储储 e e 44 duces a two moment Landau damping model. Note that the Landau damping operator ( 兩 k 储兩 ) acts on an odd moment (u 储 ), which has no equilibrium component, so that there is no linear magnetic flutter contribution to the Landau closure, avoiding a concern expressed by Finn and Gerwin.54 However, magnetic flutter does introduce an additional nonlinear Landau damping term, as discussed in Ref. 47. The size of this term has been calculated in simulations and found to be small. This electron model can be viewed as an extension of the equations of Kadomtsev and Pogutse51 to include toroidal

共112兲

1 2b T , n⫺ 1⫹b/2 i 共 2⫹b 兲 2 ⬜ i

共113兲

where b⫽k⬜2 ␳ 2i . This approximation is first order accurate in ¯ i →0) for b for both n i and T⬜ i , and it behaves properly (n large b. Within the gyrokinetic ordering, the parallel Ampere’s Law is33 ⵜ⬜2 A 储 ⫽⫺

␶␤ e 共 ¯u 储 i ⫺u 储 e 兲 , 2

共114兲

where ␤ e ⫽8 ␲ n 0 T 0e /B 2 . The transformation to real space is again accomplished with a Pade´ approximation ¯u 储 i ⫽

1 2b u ⫺ q . 1⫹b/2 储 i 共 2⫹b 兲 2 ⬜ i

共115兲

Poisson’s equation 关Eq. 共112兲兴 and Ampere’s Law 关Eq. 共114兲兴, together with six ion moment equations 关Eqs. 共81兲– 共86兲兴, the two electron moment equations 关Eqs. 共109兲– 共110兲兴, and the T e condition 关Eq. 共111兲兴, provide a complete description of the ten unknowns (n i , u 储 i , p 储 i , p⬜ i , q 储 i , q⬜ i ,




3213

n e , u 储 e , T e , ␾, and A 储 ). The system is solved by evolving the eight partial differential equations in time, while using Eq. 共112兲 to solve for ␾, Eq. 共114兲 to solve for u 储 e , and Eq. 共111兲 to solve for T e . It is shown in Sec. II C of Ref. 55 that this system of equations exactly reproduces the kinetic dispersion relation in the local fluid limit (k 2储 v 2ti Ⰶ ␻ 2 Ⰶk 2储 v 2te , 兩 ␻ d 兩 Ⰶ 兩 ␻ 兩 , k⬜2 ␳ 2i Ⰶ1). The ability of the equations to model nonlocal kinetic toroidal drift instabilities is tested in the following section with a series of linear benchmarks.

VI. LINEAR BENCHMARKS WITH KINETIC THEORY

Benchmarking the model against linear kinetic theory is an important step in verifying the accuracy and reliability of both the electromagnetic gyrofluid physics model and the simulation code used to implement the model. An extensive series of linear benchmarks in the electrostatic case is given in Ref. 30, so we focus here on the impact of finite plasma ␤. Finite-␤ effects on the collisionless ion temperature gradient 共ITG兲 instability are benchmarked in toroidal flux tube geometry. In addition, the growth rates and real frequencies of the kinetic ballooning mode 共KBM兲 are benchmarked in toroidal geometry. Both the case with no temperature gradient and the more interesting case with finite ion temperature gradient are investigated. It is shown that the gyrofluid model is able to reproduce the finite growth rates of the KBM below the ideal MHD ␤-limit in this case. It is important to note that this set of benchmarks provides a test of the electron physics model, as well as the ion physics model. While a simple adiabatic electron model can produce the correct ITG growth rate in the electrostatic limit, this is not the case for the finite-␤ ITG and KBM modes considered here, as discussed for example in Sec. II C of Ref. 55. A description of electron ⵜB and curvature drift motion and proper consideration of magnetic flutter across equilibrium electron temperature gradients are required to accurately calculate growth rates of both the finite-␤ ITG and KBM instabilities. A. The finite-␤ ITG instability

The toroidal ion temperature gradient 共ITG兲 instability is widely thought to play an important role in core transport. Capturing the finite-␤ effects on this mode has been a principal motivation for developing an electromagnetic turbulence model. Linear kinetic theory for the electromagnetic case in nonlocal toroidal geometry is quite involved, and a fairly limited set of codes is available. A code developed by Kim, Horton, and Dong,16 solves a simplified set of integral equations in ballooning coordinates, using an s⫺ ␣ equilibrium model. Figure 1 shows a benchmark using parameters selected from Fig. 6共a兲 in Ref. 16. The plot shows linear growth rate vs the safety factor q, at two values of ␤. Quantitative agreement in the finite-␤ case is found to be as good as in the electrostatic case. The trend emphasized in Ref. 16, that finite-␤ effects become more important at higher q, is reproduced by the gyrofluid model.

FIG. 1. Linear growth rates of the toroidal ITG mode as a function of the safety factor q, for ␤ ⫽0 and ␤ ⫽0.8%, with ␩ i ⫽2.5, ␩ e ⫽2, k ␪ ␳ i ⫽0.5, ⑀ n ⫽0.2, s⫽0.6, and ␶ ⫽1. The gyrofluid model is compared to linear kinetic theory in sˆ ⫺ ␣ geometry, with ␣ ⫽q 2 ␤ e / ⑀ n 关 1⫹ ␩ e ⫹ ␶ (1⫹ ␩ i ) 兴 chosen to be consistent with ␤ and q.

The structure of the eigenfunctions of ␾ and A 储 in ballooning space has also been analyzed. For the parameter set ␤ ⫽0.8%, ␩ i ⫽2.5, ␩ e ⫽2, k ␪ ␳ i ⫽0.5, ⑀ n ⫽0.2, s⫽0.6, q ⫽1.5, and ␶ ⫽1, the gyrofluid eigenfunctions have been compared to Fig. 5 of Ref. 16. Good agreement is found in both the shape and parity of the real and imaginary eigenfunctions of ␾ and A 储 as well as in the ratio A 储 / ␾ Ⰶ1. We note that the real part of ␾ has even parity, while the real part of A 储 is odd, and in the normalized units, the ratio ␾ max /A储max⯝15. The eigenfunctions extend roughly 2 ␲ in ballooning angle before becoming negligible. The shape and parity of these eigenfunctions and the ratio A 储 / ␾ Ⰶ1 are all typical of the finite-␤ ITG mode. A second set of toroidal benchmarks employing the widely used GS2 linear gyrokinetic code developed by Kotschenreuther56 is given in Ref. 32. Good agreement is found in the growth rate and frequency spectra of the finite-␤ ITG mode.

B. The kinetic ballooning mode

The electromagnetic gyrofluid model also introduces instabilities in the shear Alfveń branch of the dispersion relation not found in the electrostatic case. An example is the kinetic ballooning mode 共KBM兲,17–22 here defined to be an instability in the shear Alfveń branch of the dispersion relation, analogous to the ideal MHD ballooning mode, with the addition of kinetic effects such as FLR, drift resonance, and Landau damping. The KBM is driven unstable largely by bad curvature effects in the presence of density and/or temperature gradients, though kinetic effects impact the instability threshold and growth rate. 关Because the plasma equilibrium is taken to be Maxwellian, there is no fast particle drive, and hence no unstable toroidal Alfveń eigenmode 共TAE兲.兴 The KBM is expected to play an important role in transport in cases where it is driven unstable below the ideal MHD threshold by the toroidal ion drift resonance. Benchmarks are performed both in the flat temperature gradient case, where the KBM goes unstable exactly at the ideal MHD ␤ c , and the finite ion temperature gradient case, where the KBM is unstable below ␤ c .


3214


FIG. 2. Linear growth rate 共positive兲 and frequency 共negative兲 spectra of the toroidal kinetic ballooning mode. The gyrofluid model is compared to the kinetic code of Ref. 21, in a simple circular equilibrium at ␤ ⫽6.25%. Other parameters are s⫽1, q⫽2, ␶ ⫽1, ⑀ n ⫽0.25, ␩ i ⫽ ␩ e ⫽0.

1. Benchmarks with zero ion temperature gradient

A set of benchmarks is performed against the kinetic code developed by Hong, Horton, and Choi.21 It should be noted that this code does not solve the complete kinetic equations, but rather focuses on the coupling between drift and shear Alfveń waves, and neglects ion transit and bounce frequency resonant effects. Figure 2 shows a comparison with Fig. 1 in Ref. 21 关the figure captions on Figs. 1 and 2 on p. 1593 of this article have been reversed; the figure in the upper right is Fig. 1, while the figure in the lower left is Fig. 2兴. Growth rate and frequency spectra are compared in a simple circular geometry at ␤ ⫽6.25%. Good agreement is found for the frequency, which is nearly dispersionless with a phase velocity of roughly ⫺0.6c s ␳ s /L n in the ion diamagnetic direction. Agreement for the growth rate is also good, though some variance is seen at short wavelengths. A comparison of the growth rate and frequency of the KBM as a function of ␤ is given in Ref. 32, and good agreement is found. 2. Benchmarks with finite ion temperature gradient

The KBM becomes particularly interesting in the presence of finite ion temperature gradient because, as shown by Andersson and Weiland,57 finite ␩ i is a necessary and sufficient condition for instability of the shear Alfveń branch below the ideal MHD ␤ limit. Hence this mode may play a significant and direct role in driving transport in plasmas which are ideal MHD stable. A set of benchmarks is again performed, using parameters and results from Ref. 21. Figure 3 shows frequency and growth rate spectra for the toroidal KBM at two values of ␤ ⫽3.125%,6.25%. Other parameters are identical to Fig. 2, except that ␩ i ⫽2. Agreement between the two models is fairly good, with the gyrofluid model correctly accounting for the dramatic increase in growth rates at finite ␩ i . A comparison of the growth rate of the finite-␩ i toroidal KBM is given in Ref. 32, and good agreement is found, with the gyrofluid model accurately reproducing the finite growth rate of the mode both below and above the ideal ballooning ␤ limit. A final benchmark, Fig. 4, shows the growth rate dependence on the magnetic shear, for two different values of ⑀ n .


FIG. 3. Frequency 共negative兲 and linear growth rate 共positive兲 spectra for the toroidal kinetic ballooning mode in the presence of a finite ion temperature gradient. Parameters chosen are ␩ i ⫽2, ␩ e ⫽0, ⑀ n ⫽0.25, s⫽1, q⫽2, and ␶ ⫽1. The gyrofluid model is compared to a linear kinetic calculation at two values of ␤ ⫽3.125%,6.25%.

Again quantitative agreement is reasonably good, with the gyrofluid model successfully reproducing the trends emphasized in Ref. 21. VII. SUMMARY AND CONCLUSION

A model has been developed to describe electromagnetic microturbulence and transport in long mean-free-path plasmas. The model consists of a set of electromagnetic multispecies gyrofluid and electron Landau fluid equations derived by taking moments of the nonlinear toroidal electromagnetic gyrokinetic equation,4,33 along with the gyrokinetic Poisson equation and Ampere’s Law. A full hierarchy of six multispecies electromagnetic gyrofluid equations is derived, which can be used to describe both ions and electrons. However, a key result of this paper is the derivation of a reduced set of electron equations, appropriate for the efficient description of the passing electron response to turbulence on scales characteristic of ion drift and kinetic ballooning instabilities. The reduced set of electron equations is derived via an analytic expansion in temporal ( ␻ ⬃ ␻ , ␻ d ,k 储 v ti ,k 储 v A Ⰶk 储 v te ) and spatial (k⬜⫺1 ⬃ ␳ i * Ⰷ ␳ e ,c/ ␻ pe ) scales, formally carried out as an expansion in the electron–ion mass ratio, treating plasma ␤ as an order unity quantity. This expansion results in a simple set of electron fluid equations which describe electromagnetic electron dynamics on the typical ion drift and shear Alfveń length and time scales, while analytically removing the numerically

FIG. 4. Linear growth rate of the kinetic ballooning mode vs magnetic shear, at two values of ⑀ n ⫽0.1,0.25, for ␤ ⫽9.375%, k ␪ ␳ i ⫽0.3, q⫽2, ␶ ⫽1, ␩ i ⫽2, and ␩ e ⫽0. The gyrofluid model is compared to linear kinetic theory 共Ref. 21兲.



challenging electron transit time scale as well as the small electron gyroradius and skin depth length scales. While the resulting electron model is simple and relatively straightforward to implement numerically, it describes substantial physics not incorporated in the adiabatic electron models that have been used to describe the passing electrons in many previous gyrofluid and gyrokinetic simulations. In addition to finite-␤ effects and Alfveń wave dynamics, the model also incorporates electron E⫻B, curvature, and ⵜB drift motion, as well as the E⫻B nonlinearity and nonlinear terms due to magnetic flutter. The use of an electron temperature closure appropriate for ␻ ⬃ ␻ A Ⰶk 储 v te allows for the proper inclusion of the ⵜT e as well as the ⵜn e drive of the kinetic ballooning mode. In the lowest order form, including no finite electron mass effects, the reduced electron equations lead to an electron response in which the total, equilibrium plus fluctuating, electron temperature is constant along the total magnetic ˜ )•ⵜ(T 0e ⫹T ˜储 )⫽0. This intuitive result is exfield, (B0⫹B e pected from our ordering of the velocities ␻ /k⬃ v ti , v A Ⰶ v te , implying the characteristic time scale of both plasma and magnetic fluctuations is long compared to the time scale on which the electrons thermalize. It is important to note that this isothermal response is quite different from the occasionally employed closures ⵜ 储 ˜T e ⫽0, or ˜T e ⫽0, both of which fail to properly account for magnetic fluctuations across the equilibrium temperature gradient. In the finite m e version of the reduced electron model, formally carried to order (m e /m i ) 1/2, models of parallel electron Landau damping as well as electron–ion collisions enter, and the isothermal electron condition is only approximate. Ion species are described with the full set of six gyrofluid moment equations, truncated with kinetic closures based on Refs. 12, 30, and 44, incorporating both parallel and toroidal kinetic effects. The full set of electromagnetic ion gyrofluid equations include models of parallel Landau damping, ion drift resonance, ion–ion collisions, and linear and nonlinear finite-Larmor-radius 共FLR兲 effects. Magnetic fluctuations enter the ion equations through the inductive electric field, as well as through several linear and nonlinear magnetic flutter terms. The model has been benchmarked with linear gyrokinetic calculations, and good agreement has been found for the growth rates and real frequencies of both the finite-␤ toroidal ion temperature gradient 共ITG兲 and kinetic Alfveń ballooning 共KBM兲 instabilities. In particular, the model has been found to accurately reproduce the finite-␤ stabilization of the toroidal ITG mode. The model is also able to reproduce the behavior described by Refs. 21, 22, and 57, in which the kinetic ballooning mode is driven unstable below the ideal MHD ballooning limit ( ␤ c ) by ion drift resonance. The model with reduced electron equations is intended for use in nonlinear simulations of ion drift/shear Alfveń turbulence and transport. Use of the full six moment model for the electrons should allow simulation of turbulence on the smaller scales characteristic of electron drift instabilities. Considerations associated with the use of linearized kinetic closures in nonlinear simulations have been discussed exten-


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sively in the gyrofluid literature.10–14,30,31,39 An important strength of the reduced electron equation derivation given here is that it requires no separate closure approximations for nonlinear terms in the limit m e /m i →0. One important caveat for the multispecies gyrofluid equations is that the closures employed here reproduce the fast linear collisionless damping of zonal flows for t⬍qR/ v ti 冑⑀ , but do not account for the residual undamped component of the zonal flow noted by Rosenbluth and Hinton,48 which can be important at low collisionality. Efforts to incorporate neoclassical effects such as this residual flow into the toroidal closure are ongoing.49 Nonlinear Landau damping processes may also be important, particularly in weak turbulence/low-collisionality regimes where the rate of scattering out of the resonant region is small. Important avenues for future work include nonlinear benchmarking with gyrokinetic codes to assess the impact of nonlinear kinetic effects, and incorporation of an appropriate model for trapped electrons. Simulations of nonlinear toroidal microturbulence using the reduced electron model have been carried out, and are described in Refs. 32 and 55. ACKNOWLEDGMENTS

This work was supported by the U.S. Department of Energy under Grant No. DE-FG03-95ER54309 and Contract No. DE-AC02-76CH03073, and by a NSF fellowship. The authors wish to thank M. A. Beer and W. Dorland for many useful discussions, for providing the massively parallel, electrostatic toroidal gyrofluid code that was the starting point of our calculations, and for help with the GS2 code. They also thank M. Kotschenreuther for helpful discussions and for the use of his gyrokinetic GS2 code. Computer resources were provided by the National Energy Research Supercomputer Center, and by the Los Alamos Advanced Computing Lab, as part of the computational grand challenge Numerical Tokamak Turbulence Project. E. A. Frieman and L. Chen, Phys. Fluids 25, 502 共1982兲. W. W. Lee, Phys. Fluids 26, 556 共1983兲. 3 D. H. E. Dubin, J. A. Krommes, C. Oberman, and W. W. Lee, Phys. Fluids 26, 3524 共1983兲. 4 T. S. Hahm, W. W. Lee, and A. Brizard, Phys. Fluids 31, 1940 共1988兲. 5 F. Jenko and B. D. Scott, Phys. Plasmas 6, 2705 共1999兲. 6 W. Dorland, F. Jenko, M. Kotschenreuther, and B. N. Rogers, Phys. Rev. Lett. 85, 5579 共2000兲; B. N. Rogers, W. Dorland, and M. Kotschenreuther, ibid. 85, 5336 共2000兲. 7 W. W. Lee, J. Comput. Phys. 72, 243 共1987兲. 8 S. E. Parker, W. W. Lee, and R. A. Santoro, Phys. Rev. Lett. 71, 2042 共1993兲. 9 A. M. Dimits, J. A. Byers, T. J. Williams, B. I. Cohen, X. Q. Xu, R. H. Cohen, J. A. Crotinger, and A. E. Shestakov, in Plasma Physics and Controlled Nuclear Fusion Research, 1994 共International Atomic Energy Agency, Vienna, 1994兲, Vol. 2, pp. 457–462. 10 R. E. Waltz, R. R. Dominguez, and G. W. Hammett, Phys. Fluids B 4, 3138 共1992兲. 11 M. A. Beer, G. W. Hammett, W. D. Dorland, and S. C. Cowley, Bull. Am. Phys. Soc. 37, 1478 共1992兲. 12 W. Dorland, ‘‘Gyrofluid models of plasma turbulence,’’ Ph.D. thesis, Princeton University, 1993. 13 G. W. Hammett, M. A. Beer, W. Dorland, S. C. Cowley, and S. A. Smith, Plasma Phys. Controlled Fusion 35, 973 共1993兲. 14 R. E. Waltz, G. D. Kerbel, and J. Milovich, Phys. Plasmas 1, 2229 共1994兲. 15 J. Q. Dong, W. Horton, and J. Y. Kim, Phys. Fluids B 4, 1867 共1992兲. 16 J. Y. Kim, W. Horton, and J. Q. Dong, Phys. Fluids B 5, 4030 共1993兲. 1

2


3216 17


W. M. Tang, J. W. Connor, and R. J. Hastie, Nucl. Fusion 20, 1439 共1980兲. 18 C. Z. Cheng, Phys. Fluids B 25, 1020 共1982兲. 19 T. S. Hahm and L. Chen, Phys. Fluids 28, 3061 共1985兲. 20 M. Kotschenreuther, Phys. Fluids 29, 2898 共1986兲. 21 B.-G. Hong, W. Horton, and D.-I. Choi, Phys. Fluids B 1, 1589 共1989兲. 22 F. Zonca, L. Chen, and R. Santoro, Plasma Phys. Controlled Fusion 38, 2011 共1996兲. 23 B. N. Rogers and J. F. Drake, Phys. Rev. Lett. 79, 229 共1997兲. 24 A. Das, P. Diamond, M. Rosenbluth, and A. Smolyakov, Bull. Am. Phys. Soc. 44, 1263 共1999兲. 25 S. I. Braginskii, in Reviews of Plasma Physics, edited by M. A. Leontovich 共Consultants Bureau, New York, 1965兲, Vol. 1, pp. 205–311. 26 B. Scott, Plasma Phys. Controlled Fusion 39, 1635 共1997兲. 27 A. Zeiler, J. F. Drake, and B. N. Rogers, Phys. Rev. Lett. 84, 99 共2000兲. 28 A. Zeiler, D. Biskamp, J. F. Drake, and B. N. Rogers, Phys. Plasmas 5, 2654 共1998兲. 29 X. Q. Xu, R. H. Cohen, G. D. Porter et al., Nucl. Fusion 40, 731 共2000兲. 30 M. A. Beer, ‘‘Gyrofluid models of turbulent transport in tokamaks,’’ Ph.D. thesis, Princeton University, 1995. 31 M. A. Beer and G. W. Hammett, Phys. Plasmas 3, 4046 共1996兲. 32 P. B. Snyder and G. W. Hammett, Phys. Plasmas 8, 744 共2001兲. 33 A. Brizard, Phys. Fluids B 4, 1213 共1992兲. 34 A. Brizard, J. Plasma Phys. 41, 541 共1988兲. 35 P. J. Catto and K. T. Tsang, Phys. Fluids 20, 396 共1977兲; T. M. Antonsen and B. Lane, ibid. 23, 1205 共1980兲. 36 The use of this ␤ Ⰶ1 approximation may seem inconsistent with the goal of including finite ␤ effects. However, it allows treatment of ␤ values up to the MHD critical ␤ c for all except very low aspect ratio cases, where ␤ c ⬃1. The critical point is that while the electrostatic approximation requires ␤ Ⰶ ␤ c , the above approximations require only that ␤ Ⰶ1. That is, our equations are valid for ␤ ⬃ ␤ c Ⰶ1, and can be extended to ␤ ⬃1 by adding ␦ B 储 and separately treating the curvature and ⵜB drifts. 37 H. Berk and R. Dominguez, J. Plasma Phys. 18, 31 共1977兲.

P. B. Snyder and G. W. Hammett T. S. Hahm, Phys. Fluids 31, 2670 共1988兲. W. Dorland and G. W. Hammett, Phys. Fluids B 5, 812 共1993兲. 40 R. Linsker, Phys. Fluids 24, 1485 共1981兲. 41 G. S. Lee and P. H. Diamond, Phys. Fluids 29, 3291 共1986兲. 42 R. E. Waltz, Phys. Fluids 31, 1962 共1988兲. 43 S. Hamaguchi and W. Horton, Plasma Phys. Controlled Fusion 34, 203 共1992兲. 44 G. Hammett and F. Perkins, Phys. Rev. Lett. 64, 3019 共1990兲. 45 Z. Chang and J. D. Callen, Phys. Fluids B 4, 1167 共1992兲. 46 M. Medvedev and P. Diamond, Phys. Plasmas 3, 863 共1996兲. 47 P. B. Snyder, G. W. Hammett, and W. Dorland, Phys. Plasmas 4, 3974 共1997兲. 48 M. N. Rosenbluth and F. L. Hinton, Phys. Rev. Lett. 80, 724 共1998兲. 49 M. Beer and G. Hammett, Bull. Am. Phys. Soc. 44, 1265 共1999兲. 50 E. Gross and M. Krook, Phys. Rev. 102, 593 共1956兲. 51 B. B. Kadomtsev and O. P. Pogutse, in Plasma Physics and Controlled Nuclear Fusion Research, 1984 共International Atomic Energy Agency, Vienna, 1985兲, Vol. 2, p. 69. 52 Normalization to the Alfveń scales v A and m i , or the sound wave scales c s and m i is equivalent. We also note that the reduced electron equations can be derived directly from a mass ratio expansion of Eqs. 共81兲–共86兲, provided that the presence of the electron mass in the normalization 关Eq. 共67兲兴 is properly taken into account. 53 S. C. Cowley, ‘‘Some aspects of anomalous transport in tokamaks: Stochastic magnetic fields, tearing modes, and nonlinear ballooning instabilities,’’ Ph.D. thesis, Princeton University, 1985. 54 J. Finn and R. Gerwin, Phys. Plasmas 3, 2469 共1996兲. 55 P. B. Snyder, ‘‘Gyrofluid theory and simulation of electromagnetic turbulence and transport in tokamak plasmas,’’ Ph.D. thesis, Princeton University, 1999. 56 M. Kotschenreuther, G. Rewoldt, and W. M. Tang, Comput. Phys. Commun. 88, 128 共1995兲. 57 P. Andersson and J. Weiland, Phys. Fluids 31, 359 共1988兲. 38 39


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